1. Introduction: Theoretical Developments; From Unconscious Inference to Intrinsic Image:

1.1. Basic ambiguity: luminance vs. lightness; lightness constancy

1.2. Inferring the illuminance: Helmholtz

1.3. Hering’s paradox

1.4. Gestalt and the relational approach

1.5. Contrast theories versus relational theories

1.6. Intrinsic image theories

1.6.1. Edge integration.

1.6.2. Edge classification.

1.6.3. Parsing into layers

1.8. Two weaknesses of the intrinsic image models.

1.8.1. The problem of errors.

1.8.2. The missing anchor.

2. The Anchoring Problem

2.1. A concrete example

2.2. Mapping luminance onto lightness

2.3. Wallach on anchoring: Highest Luminance Rule

2.4. Helson on anchoring: Average Luminance Rule

2.5. Anchoring in intrinsic image models

2.5.1. Anchoring applied to reflectance layer.

2.6. Absolute lightness versus absolute luminance.

2.7. Anchoring versus scaling

3. The Rules of Anchoring in Simple Displays

3.1. What are minimal conditions?

3.2. Highest Luminance versus Average Luminance

3.3. Luminosity Problem: Direct Contradiction to HLR

3.3.1. Wallach on increments: HLR does not apply

3.3.2. Common factor for decrements and increments: Geometric, not photometric

4. Area effects

4.1. Surround rule fails

4.2. Highest luminance plus area

4.3. The Area Rule

4.3.1. Other empirical findings

4.4. Luminosity and the Area Rule

4.4.1. Area Rule implies luminosity:

4.4.2. Luminosity induction versus grayness induction.

4.4.3. Phenomena related to the Area Rule

4.4.4. Luminosity threshold and area: Bonato and Gilchrist

4.4.5. Maximum luminance is not the same as the anchor

4.5. Anchoring rules for simple images: A summary

5. Anchoring in Complex Images: A New Theory

5.1. What are the relevant components of a complex image?

5.2. Framework

5.3. Local and global frameworks

5.4. Weighting

5.5. Belongingness and grouping factors

5.5.1. Importance of T-junctions

5.5.1. Role of luminance gradients

5.6. Theoretical value of belongingness.

5.7. The scale normalization effect.

6. Testing the model: The staircase Gelb effect

6.1. Applying the anchoring model

6.2. Compromise

6.3. Weighting factors in the staircase Gelb effect

6.3.1. Configuration.

6.3.2. Articulation.

6.3.3. Field Size.

6.3.4. Perceived size is crucial, not retinal size

6.3.5. Insulation

7. Anchoring and the Pattern of Lightness Errors

7.1. Gilchrist (1988): Failures due to classification problem

7.1.2. Staircase Gelb data inconsistent

7.1.3. Anchoring model consistent with Gilchrist (1988) data.

7.2. Source of Errors

8. The Model Applied to Illumination-Dependent Errors

8.1. Errors associated with level of illumination

8.2. Errors associated with reflectance of target.

8.3. Errors associated with reflectance of backgrounds

8.3.1. The key: Increments versus decrements

8.4. Errors associated with articulation level

8.4.1. Burzlaff experiments.

8.4.2. Arend & Goldstein experiments.

8.4.3. Schirillo experiments.

8.4.4. Application of model to depth and lightness.

8.5. Errors associated with field size

9. The Model Applied to Background-Dependent Errors (Simultaneous Lightness Contrast)

9.1. Anchoring component

9.2. The scale normalization component

9.3. Predictions of the model

9.3.1. The locus of the error.

9.3.2. The size of the error.

9.3.3. No contrast effect with double increments

9.3.4. Segregating effect of luminance ramps

9.3.5. Belongingness creates the illusion

9.4. The Benary effect

9.5. White’s illusion.

9.6. Checkerboard contrast.

9.7. Agostini and Proffitt: Common fate.

9.8. Laurinen and Olzak.

9.9. Wolff illusion

9.10. Depth manipulations:

9.11. Adelson’s corrugated Mondrian

9.11.1. Wishart experiments.

10. Brightness Induction: Contrast or Anchoring?

10.1. Induction

10.2. Area effects

10.3. Separation

10.4. Depth separation: Gogel and Mershon

10.5. Gelb effect: Contrast versus anchoring

10.6. Summary: application of model to reduction conditions.

10.6.1. Lightness versus brightness

11. Tests of scaling normalization effect

11.1. Crispening effect

12. Problems for the Intrinsic Image Models

12.1. Problem of the staircase Gelb effect data.

12.2. Problem of area effects.

12.3. Problem of articulation effects.

12.4. Problem of errors.

12.5. A response paradox.

13. New Model vs. Intrinsic Image Models

13.1. Classified edge integration

13.2. A Critical Test

13.2.1. Method and results: New model wins.

13.2.2. Low articulation replication.

13.2.3. Review of 1983 data.

14. Evaluating The Model

14.1. Range of application.

14.2. Prediction of errors.

14.3. Unification of constancy failures.

14.4. Rigor and concreteness.

14.5. The role of perceived illumination.

14.6. Choice of scaling rule for the global framework.

15. Debt to the Early Literature

15.1. Koffka

15.2. Compromise and intelligence

15.3. The aperture problem for lightness

An Anchoring Theory of Lightness Perception

1Gilchrist, A., 2Kossyfidis, C., 3Bonato, F., 4Agostini, T., 3Cataliotti, J.,

1Li, X., 5Spehar, B., 1Szura, J.

1Rutgers University

2Deceased

3St. Peters College

4University of Trieste

5University of New South Wales

This paper is dedicated to the memory of Christos Kossyfidis.

Running Head: AN ANCHORING THEORY OF LIGHTNESS PERCEPTION

Abstract

A review of the field of lightness perception from Helmholtz to the present shows the most adequatetheories of lightness perception to be the intrinsic image models. Nevertheless these models fail on twoimportant counts: they contain no anchoring rule and they fail to account for the pattern of errors insurface lightness. Recent work on both the anchoring problem and the problem of errors has produced anew model of lightness perception, one that is qualitatively different from the intrinsic image models.The new model, which is based on a combination of local and global anchoring of lightness values,appears to provide an unprecedented account of a wide range of empirical results, both classical andrecent, especially the pattern of errors. It provides a unified account of both illumination-dependentfailures of constancy and background-dependent failures of constancy, resolving a number of

long-standing puzzles.

1. Introduction: Theoretical Developments; From Unconscious Inference to IntrinsicImage:

1.1. Basic ambiguity: luminance vs. lightness; lightness constancy

We present here a new theory of how the visual system assigns lightness values, or perceived black,white, or gray values to various regions of the retinal image. The theory grew out of two problemsfacing what are probably the most advanced models of lightness perception: the intrinsic image models.

To understand the problem of lightness constancy, it is necessary to understand the ambiguousrelationship between luminance values (light intensities) within the retinal image, and the lightnessvalues of the surfaces in our perceived world. A key problem is that luminance values in the retinalimage are a product, not only of the actual physical shade of gray of the imaged surfaces, but also andeven more so, of the intensity of the light illuminating those surfaces. The luminance of any region ofthe retinal image can vary by a factor of no more than thirty to one, as a function of the physicalreflectance of that surface. But it can vary as a factor of a billion to one as a function of the amount ofillumination on that surface. The net result is that any given luminance value can be perceived asliterally any shade of gray, depending on its context within the image. Despite this challenge, weperceive shades of surface grays with rough accuracy. This is the well known problem of lightnessconstancy. Here is a brief history of attempts to solve this problem.

1.2. Inferring the illuminance: Helmholtz

In one of the earliest attempts to solve this problem, Helmholtz (1866) proposed that the luminance of aregion in the retinal image is compared with the perceived intensity of the illumination in that region ofthe visual scene. Dividing the luminance by the illumination yields reflectance and this is how aphysicist might determine the reflectance of a surface. But as an account of human lightness, the theoryhas been fraught with both logical and empirical difficulties.

1.3. Hering’s paradox

Hering (1874) argued that Helmholtz’ position is illogical because, given the luminance at any point,one would need to know the surface reflectance to infer the illumination. But one would also need toknow the illumination to infer the surface reflectance. Hering emphasized the role of sensorymechanisms like pupil size and adaptation. But he emphasized lateral inhibition as the main mechanism,many years before its actual discovery, thus creating the prototype of the modern contrast theories.

The publication of Katz’s (1911, 1935) book The World of Colour gave enormous momentum to theemerging field of lightness perception. "Its importance at the time of its publication can hardly beoverrated," wrote Koffka (1935, p. 241). Katz presented a thorough phenomenological analysis of thevisual experience of color. He outlined the various modes of color appearance, emphasizing especiallythe distinction between surface color and film color. Yet at the same time Katz was a rigorousexperimentalist, especially in relation to his historical context. He developed a variety of experimentalmethods for studying lightness constancy, including the now standard method involving side-by-sidefields of light and shadow. He showed that lightness constancy holds even when all three of Hering’smechanisms plus memory color are ruled out. Katz’s theoretical perspective, though initially closest to

that of Helmholtz, became strongly influenced by the Gestalt theorists.

1.4. Gestalt and the relational approach

The Gestalt theorists rejected the assumption that light per se is the stimulus for lightness, in favor ofluminance ratios or gradients. As Marr (1982, p. 259) has noted: "It is a widespread and time-honoredview, going back at least to Ernst Mach, that object color depends upon the ratios of light reflected fromthe various parts of the visual field rather than on the absolute amount of light reflected." But Koffka(1935, p. 244) was the first to base a theory of lightness perception strictly on relative luminance: "Ourtheory of whiteness constancy will be based on this characteristic of colours...that perceived qualitiesdepend upon stimulus gradients." Gelb (1929) showed that a piece of black paper appears white whenpresented alone in a spotlight, but much darker when a real white is placed next to it in the spotlight.Experiments of fundamental importance were also conducted by Kardos (1934, 1935), Burzlaff (1931),Wolff (1933, 1934), and Katona (1935), to name only a few. Reading the current literature, one wouldhardly suspect the vigorous empirical and theoretical developments that took place during the severaldecades following the publication of the first edition of Katz’s book in 1911.

Those developments were derailed by the events leading to World War II. After the war, the spotlightshifted to America, where contrast theories, spurred by the first direct physiological evidence for lateralinhibition, came to dominate the field completely. The stimulus conditions became highly reduced:luminous patches presented in dark rooms. Relative luminance came to mean contrast. The Gestaltlessons were lost in the stampede to explain lightness at the physiological level. The contrast theoristsargued that the facts of lateral inhibition rendered the vague ideas of the Gestaltists obsolete.Consequently they felt no need to cite the earlier European work, and they did not (see Gilchrist, 1996).

Several important publications in the tradition of relational determination appeared in the 1940’s, butthey were quickly assimilated to the contrast interpretation. Helson (1943, 1964), like Koffka, basedlightness values on stimulus gradients. He proposed that the luminance of a target surface is comparedwith the average luminance (in fact a weighted average) in the retinal image, such that a surface with aluminance equal to the average luminance is seen as middle gray, luminances above the average are seenas light gray or white and those below the average are seen as dark gray or black.

Wallach (1948), in a landmark paper, proposed a simple ratio theory of lightness. Presenting observerswith two disk/annulus displays, he showed that disks of different luminance appear equal in lightness aslong as the disk/annulus luminance ratios are equal.

1.5. Contrast theories versus relational theories

Hering (1874) is often described as one of the earliest to point out the importance of relative luminance.But, as Koffka (1935, p. 245) wrote about Hering: "contrast...implies an explanation not in terms ofgradient, but in terms of absolute amounts of light." This is explicit in more recent contrast theorieswithin the Hering tradition. Cornsweet (1970, p. 303) defines the correlate of perceived lightness as "thefrequency of firing of the spatially corresponding part of the visual system (after inhibition)." LikewiseJameson and Hurvich (1964) seem to acknowledge a fundamental role for relative luminance but theirposition is that the lightness values produced by a given luminance ratio in the stimulus depends, in theend, on the absolute luminance values (Jameson & Hurvich, 1961). This was the central point of theircelebrated but essentially unreplicable (Flock & Noguchi, 1970; Gilchrist & Jacobsen, 1988; Haimson,1974; Heinemann, 1988; Jacobsen & Gilchrist, 1988a and 1988b; Noguchi & Masuda, 1971) report.

Relational theories like those of Koffka, Helson, and Wallach can be driven equally well by an inputconsisting either of absolute or relative luminance values. Contrast theories, while sometimes couched inrelational terms, require absolute luminance information as well.

Although lightness perception continues to be attributed to contrast mechanisms by non-specialists,those who study surface lightness have long recognized that lateral inhibition, although crucial to theencoding of stimulus values, plays no more substantial role.

1.6. Intrinsic image theories

Empirical evidence accumulated more recently (Shapley & Enroth-Cugell, 1984; Whittle & Challands,1969; Yarbus, 1967) tends to support the idea that retinal encoding processes simply encode relativeluminance (see Gilchrist, 1994). And yet there has been a reluctance to embrace such a simple idea.

In the past two decades, more effective models of lightness perception have emerged. They haveevolved, not out of contrast theories, but out of attempts to correct several limitations in Wallach’ssimple ratio formula. Although it has become widely agreed that the concept of luminance ratios atedges goes far toward explaining the traditional problem of lightness constancy, this same insight hasrevealed a second constancy problem. When the same piece of gray paper is viewed successively againstdifferent backgrounds, the luminance ratio at the edge of the paper changes dramatically, yet the paperappears to change very little in lightness, contrary to the ratio principle. This constancy of lightness withrespect to changing background has been labeled Type II constancy, to distinguish it from lightnessconstancy with respect to changing illumination, or Type I constancy. Ross and Pessoa (personalcommunication) have proposed the more memorable terms illumination-independent constancy (Type I)and background-independent constancy (Type II) and we have adopted that usage. A third kind might becalled veil-independent constancy (Gilchrist & Jacobsen, 1983).

1.6.1. Edge integration.

Wallach’s ratio rule deals effectively with adjacent luminances. But background-independent constancyseems to require a mechanism by which the luminance values of widely separated regions in the retinalimage can be compared. The recognition of this need led to several papers, which appeared at almost thesame time. Land and McCann (1971); Arend, Buehler, and Lockhead (1971); and Whittle and Challands(1969) offered evidence and arguments suggesting that the visual system is capable of deriving theluminance ratio between two surfaces remote from each other in the image. The exact mechanism forthis is unknown, but one suggestion is that luminance ratios at every edge encountered along an arbitrarypath from one surface to its remote pair are mathematically integrated.

1.6.2. Edge classification.

Gilchrist (1977, 1979, 1980) demonstrated empirically an observation that had been made by Koffka in1935 (p. 248) that "not all gradients are equally effective as regards the appearance of a particular fieldpart....," and, in another passage (p. 246): "Clearly two parts at the same apparent distance will, ceterisparibus, belong more closely together than field parts organized in different planes." Gilchrist found thatperceived lightness values depend primarily on luminance ratios between adjacent regions perceived tolie in the same plane, as opposed to luminance ratios between any two adjacent parts of the visual field.

Inspired by Koffka’s observation that some luminance ratios are relatively effective in determining

surface lightness while others are relatively ineffective, Gilchrist (1977; 1979; Gilchrist, Delman, &Jacobsen, 1983) proposed a distinction between what he called reflectance edges and illuminance edges.Reflectance edges are those luminance borders in the retinal image that are caused by a change in thereflectance (or pigment) of the surface being viewed while illuminance edges are those that are causedby changes in the intensity of the illumination on a surface, such as the border of a cast shadow, forexample, or the luminance step at a corner. Gilchrist proposed that the visual system must classify edgesin the image into one of these two main categories, before edge integration. Then an integration of allthe edges in the reflectance category can yield a map of all the reflectances in the visual field, just as anintegration of all the edges in the illuminance class yields a map of the illuminance pattern within thevisual field. In effect, Gilchrist proposed to use edge classification as a wedge to pry the retinal imageinto two overlapping layers, one representing surface lightness values, the other representing the patternof illuminance on those surfaces.

1.6.3. Parsing into layers

At the same time, Bergström (1977) proposed that luminance variations within the retinal image arevector analyzed into three components: one for surface reflectance, one for illumination, and one forthree-dimensional form. Bergström’s distinction between illumination changes and three-dimensionalform changes is roughly equivalent to Gilchrist’s further breakdown of illumination edges into castilluminance edges and attached illuminance edges.

Adelson and Pentland (1990) have recently proposed an elegant scheme that bears a strikingresemblance to Bergström’s model. They liken the visual system to a three person workshop crew thatproduces theatrical sets. One person is a painter, one is an illumination expert, and one bends metal. Anyluminance pattern can be produced by any of the three specialists. The painter can paint the pattern. Thelighting expert can produce the pattern with variations in the illumination. And the metalworker cancreate the pattern by shaping the surface, as in shape from shading. But the cost of these three methodsis not the same, and this sets up an economy principle in which each desired pattern is to be created inthe cheapest possible way, reminiscent of the Gestalt simplicity principle (Gerbino, 1994).

In 1978, Barrow and Tenenbaum proposed that every retinal image is composed of a set of what theycalled intrinsic images. One intrinsic image would contain the array of reflectance values in the scene, asecond, the array of illumination intensities, a third, the array of depth values, and so on.

The Bergström, Gilchrist, Barrow and Tenenbaum, and Adelson and Pentland models all have incommon the idea that the retinal image is parsed into a set of overlapping layers, much as in the scissionidea made popular by Metelli (1985; Metelli, et al, 1985) in his analysis of perceived transparency. Wewill refer to these models as intrinsic image models. An excellent discussion of them can be found in arecent chapter by Arend (1994).

1.8. Two weaknesses of the intrinsic image models.

Intrinsic image models are the most advanced models in the continuing development of lightness theory.They offer an explanation of both illumination-independent and background-independent constancy. Yetthey are incomplete in two very important, though different, ways: (1) they cannot account for errors inlightness perception and (2) they have no anchoring rule. We consider these in turn.

1.8.1. The problem of errors.

The goal of the computational enterprise that produced the intrinsic image models has been themodeling of a completely veridical lightness perception system. This is implied in terms like "inverseoptics" and "recovering reflectance." In that sense it has been consistent with the goals of machinevision. Failures of constancy and other perceptual errors have been largely ignored. Even in humanvision there is a good reason for this approach. The achievement of constancy and veridicality in theperception of surface lightness is stunning, especially when the various challenges to constancy arerecognized. This degree of veridicality does not happen by accident; it cannot be merely the byproductof a system with goals other than veridicality. Somehow a very robust truth-seeking quality must lie atthe heart of the system. Thus if the achievement of veridicality is considered to be more impressive thanthe degree of failure, it makes sense to begin by trying to model a system that comes as close to veridicalperception as possible.

Nevertheless, unlike the situation in machine vision, a theory of human lightness must include anaccount of errors. To the extent that veridicality fails in lightness perception and to the extent that theintrinsic image models predict veridicality, to that extent the models must fail to account for humanlightness perception. But perhaps more importantly, a systematic analysis of human lightness errors canbe a powerful tool for revealing how surface lightness is processed by the human visual system. There isa simple and compelling logic behind the study of errors:

1. Errors in lightness perception are always present, however slight.

2. These errors are systematic, not random.

3. The pattern of errors must reflect visual processing. This pattern is the signature of the visualsystem.

The theoretical picture would be brighter if the empirical pattern of errors could be produced bytweaking the veridicality models. But no such prospect is in sight. For example there is no coherentapproach that can explain both illumination-dependent failures of constancy (Type 1), andbackground-dependent failures of constancy (Type 2), despite an attempt by Gilchrist (1988).Background-dependent failures of constancy, of which the textbook version of simultaneous lightnesscontrast is the best-known example, are typically described as simply reflecting the operation of themechanism that achieves illumination-independent constancy. But how are illumination-dependentfailures of constancy to be explained? Is there no systematic relationship between these two kinds offailure? We consider a unified account of illumination-dependent and background-dependent failures tobe a major goal of a theory of lightness errors.

1.8.2. The missing anchor.

The second shortcoming in the intrinsic image models, in their current form, is that they are missing anessential component for veridical perception: an anchoring rule. We now turn to an extended discussionof the anchoring problem, which in turn will bring us back to the errors problem.

We will find that the attempt to fill a crucial gap in the intrinsic image approach by finding the missinganchoring rule in fact turns out to undermine the entire intrinsic image approach. At the same time itprovides the outlines of a very different approach to surface lightness, one that excels in its ability toexplain the pattern of errors found across a very broad range empirical results.

2. The Anchoring Problem

2.1. A concrete example

Although the ambiguous relationship between the luminance of a surface and its perceived lightness iswidely understood, there has been little appreciation of the fact the relative luminance values arescarcely less ambiguous than absolute luminance values. For instance, consider a pair of adjacentregions in the retinal image whose luminance values stand in a five to one ratio (see Figure 1). This fiveto one ratio informs the visual system only about the relative lightness values of the two surfaces, nottheir specific or absolute lightness values. It informs only about the distance between the two grayshades on the phenomenal gray scale, not the specific location of either on that scale. There is an infinitefamily of pairs of gray shades that are consistent with the five to one ratio. For example, if the fiverepresents white then the one represents middle gray. But the five might represent middle gray, in whichcase the one will represent black. Indeed it is even possible that the one represents white and the fiverepresents an adjacent self-luminous region. So the solution is not even restricted to the scale of surfacegrays.

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The anchoring problem is the problem of how the visual system ties relative luminance values extractedfrom the retinal image to specific values of perceived black, white, and gray.

2.2. Mapping luminance onto lightness

To derive specific shades of gray from relative luminance values in the image, one needs an anchoringrule. An anchoring rule defines at least one point of contact between luminance values in the image andgray scale values along our phenomenal black to white scale. Lightness values cannot be tied to absoluteluminance values because there is no systematic relationship between absolute luminance and surfacereflectance, as noted earlier. Rather, lightness values must be tied to some measure of relativeluminance.

The anchoring problem, closely related to the issue of normalization (Horn, 1977, 1986), is illustrated inFigure 2. Consider the following remark by Koffka (1935, p. 255) in relation to Figure 2: "...the stimulusgradient...alone does not determine the absolute position of this apparent gradient.... This wholemanifold of colours may be considered as a fixed scale, on which the two colours produced by the twostimulations.... keeping the same distance from each other, may slide, according to the generalconditions. I have called this the principle of the shift of level." Anchoring concerns where this slidingrelationship comes to rest. Koffka suggests here that the relative lightness of two regions in an imagecan remain fully consistent with the luminance ratio between them, even though their absolute lightness

levels depend on how the luminance values are anchored.

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2.3. Wallach on anchoring: Highest Luminance Rule

Apparently Wallach made no systematic study of the anchoring problem. He did mention in passing thatthe value of white is assigned to the highest luminance in the display and serves as the standard fordarker surfaces (Wallach, 1976, p. 8). Land, McCann, and Horn (Horn, 1977; Land & McCann, 1971;McCann, 1987, 1994) have also adopted this rule, which we will call the Highest Luminance Rule.

2.4. Helson on anchoring: Average Luminance Rule

Although Helson neither discussed the anchoring problem per se, nor made any systematic investigationof it, his entire adaptation-level theory is built, in effect, on an anchoring rule. His rule is that theaverage luminance in the visual field is perceived as middle gray, and this value serves as the standardfor both lighter and darker values. This average luminance rule, which is closely related to the grayworld hypothesis (Hurlbert, 1986) has found its way into more recent models (Buchsbaum, 1980),especially in the chromatic domain. It is implicit in the concept of equivalent background (Bruno, 1992;Bruno, Bernardis, & Schirillo, in press; Brown, 1994; Schirillo & Shevell, 1996). Land (1983) revertedto the average luminance rule in a later version of Retinex theory.

No one has in fact proposed a rule whereby the lowest luminance in the field is seen as black, but inprinciple such a rule is possible.

2.5. Anchoring in intrinsic image models

2.5.1. Anchoring applied to reflectance layer.

None of the intrinsic image models contains any specific anchoring rule. However the task is simplifiedbecause the various factors that can modulate luminance within the image, that is reflectance,illuminance, and three dimensional form, have already been segregated into separate layers. The obviousnext step would be to apply something like the Highest Luminance Rule solely to the reflectanceintrinsic image.

2.6. Absolute lightness versus absolute luminance.

The issue of relative versus absolute lightness, central to the anchoring problem, is independent of theissue of whether relative or absolute luminance values are encoded at input. The former applies to theoutput, the latter applies to the input. Normalization typically refers to the fact that absolute luminance

values are lost in the encoding of retinal luminance contrasts, whereas the anchoring problem concernslightness values, not luminance values. It should be noted that even if absolute luminance values wereencoded at input (a dubious assumption; see Shapley and Enroth-Cugell, 1984; Whittle & Challands,1969) or somehow recovered later (perhaps by combining relative luminance information with adaptivestate information), the anchoring problem would remain completely unsolved, due to the lack ofcorrelation between luminance and lightness.

2.7. Anchoring versus scaling

Anchoring concerns the mapping of relative luminance values from the image onto perceived shades ofsurface gray. A parallel aspect of this mapping, that we call the scaling problem, is illustrated in Figure3. It is distinct from the anchoring problem, though possibly less important. It concerns how the rangeof luminances in the image is mapped onto a range of perceived grays. Re-scaling can take one of twoforms: compression, in which the range of perceived surface grays is less (on a log scale) than the rangeof luminances in the image; and expansion (Brown & MacLeod, 1992, use the term gamut expansion),which is just the reverse.

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While anchoring can be said to concern the constancy of absolute shades of gray, scaling can be said toconcern the constancy of relative shades of gray. Wallach was more explicit about scaling than aboutanchoring. His ratio principle is a scaling rule; it asserts a one-to-one mapping from relative luminanceto relative lightness.

Contrast theories, by comparison, reject this kind of one-to-one mapping in favor of expansion. Incontrast theories, the relative luminance values encoded at input are magnified (Hurvich and Jameson,1966, p. 85), exaggerated (Leibowitz, 1965, p. 57), or amplified (Cornsweet, 1970, p. 300) by the lateralinhibitory component of the encoding process itself.

Helson, unlike Wallach, was more explicit about anchoring than about scaling. His adaptation levelrepresents an anchor; he did not specify how regions lighter and darker than the average are scaled, butwe presume that the ratio principle would apply.

3. The Rules of Anchoring in Simple Displays

Our approach will be to consider the rules of anchoring under minimum conditions for the perception ofa surface, and then to attempt to describe how the rules change as one moves systematically from simpleimages to complex images. We will find that, for simple images, anchoring depends on two dimensionsof the stimulus: relative luminance and relative area.

3.1. What are minimal conditions?

Katz (1935) and Gelb (1929) were among the first to observe that the visual perception of a surfacerequires the presence of at least two adjacent regions of non-zero luminance. Wallach (1963 p. 112)noted: "Opaque colors which deserve to be called white or gray, in other words "surface colors," willmake their appearance only when two regions of different light intensity are in contact with eachother..." Ideally these minimal conditions would be met by a pair of surfaces that fill the entire visualfield (Koffka, 1935, p. 111). One can, for instance, use the inside of a large hemisphere painted in twogray shades to cover an observer’s entire visual field. Heinemann (1971, p. 146) has expressed acommon view that the "simplest experimental arrangement for studying induction effects" consists oftwo surfaces presented within a void of total darkness. In our view, however, these conditions onlyapproximate the simplest, because there are at least three regions in the visual field, including the darksurround, instead of two. There are two borders rather than the minimum of one.

3.2. Highest Luminance versus Average Luminance

Which rule is correct under such minimal conditions, Wallach’s highest-luminance-as-white rule, orHelson’s average-luminance-as-middle-gray rule? This question cannot be answered when the stimuluscontains the entire gamut of gray shades from white to black because in that case the two rules make thesame predictions. The most direct test of these two rules is to present an observer with a truncated grayscale; an array of luminances whose range is substantially less than the 30:1 range between white andblack.

Combining a truncated gray scale with simple conditions, we placed observer’s heads inside a largeacrylic hemisphere, the inside surface of which was divided into two halves of equal size (Li andGilchrist, in press). One half was painted completely matte black; the other half middle gray. The middlegray half was seen as fully white and the black half as a darkish middle gray. All observers reported thisresult and the variability was very low. This outcome decisively favors the highest luminance rule overthe average luminance rule.

Other findings, using more complex stimuli, agree with this conclusion. In a separate line of work,Gilchrist and Cataliotti (1994) presented observers with a flat Mondrian containing 15 rectilinear piecesof gray paper, spanning only a 10:1 reflectance range (from black to middle gray). The Mondrian wasilluminated by a special projector and presented within a relatively darkened laboratory and observersindicated their perceptions by making matches from a separately housed and lighted 16-step Munsellscale. In a further experiment (Cataliotti & Gilchrist, 1995), we placed observers heads inside a smalltrapezoidal shaped empty room, all the walls of which were covered with a Mondrian pattern spanningan even smaller range (4:1). In both of these Mondrian experiments, the highest luminance wasperceived as white and no black surfaces were seen.

The results for all three tests are shown in Figure 4. Notice that in all cases the highest luminance isperceived as white, even if it is physically a dark gray surface. Notice further that the symmetry implicitin the average luminance rule is missing from these results. Although a perceived white is alwayspresent in the scene, it is not always the case that a perceived black is present in the scene, and theperceived values do not distribute themselves symmetrically about middle gray.

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The two rules have been tested indirectly in several experiments on "equivalent surrounds." Bruno(1992) asked observers to make a brightness match between two target squares of equal luminance, onesurrounded by a homogeneous region and one embedded in a checkerboard or Mondrian-like region.The luminance of the homogeneous surround that is chosen by the observers as having the same effecton a target as the checkerboard is a luminance value close to that of the highest luminance in thecheckerboard not the average. Schirillo and Shevell (1996) presented a similar pair of displays toobservers, but asked them to make a brightness match between a target square on a checkerboard andone on a uniform background of the same average luminance, as the checkerboard contrast is increasedfrom 0% to 100%. Their results for increments (regions with a luminance higher than that of thesurround) agree with those of Bruno, but their results for decrements show a different pattern (see alsoBruno, Bernardis, and Schirillo, in press).

McCann (1994) has recently shown with chromatic Mondrians that, if the average color is held constantwhile the maximum excitation in a cone channel is varied, perceived colors in the Mondrian change. Butif the maximum excitation in a cone channel is held constant while the average is varied, perceivedcolors change only slightly (see also Brown, 1994).

3.3. Luminosity Problem: Direct Contradiction to HLR

There is one phenomenon, however, that directly contradicts the highest luminance rule: the perceptionof self-luminous surfaces (Bonato & Gilchrist, 1994; Ullman, 1976). According to the highest luminancerule, white is a ceiling; nothing can appear to be brighter than white. But the fact is that we perceivevarious regions within the visual field to be self-luminous. This fact represents a serious challenge to thehighest luminance rule.

3.3.1. Wallach on increments: HLR does not apply

To be fair to Wallach (1948, 1976), his support for the highest luminance rule was neither emphatic norunambiguous. Wallach’s systematic experiments were conducted only with decrements: disk/annulusdisplays in which the disk is darker than the annulus. Under these conditions it is an empirical fact thatthe higher luminance (the annulus) always appears white, regardless of its luminance. But Wallachreported informally that when the disk is an increment, that is brighter than the surrounding annulus, itappears luminous. The implication is that the highest luminance rule applies to decrements but not toincrements. Here are his words: "We have seen that when two areas of different stimulus intensity exerttheir influence on each other, the area of lower intensity will be gray and the area of high intensity willbe white. However, the correspondence between white and gray goes no further. When the difference inintensity is varied, the value of the gray changes; the gray may even turn black. If the spot is moreintense, it will show no color other than white. An increase in the intensity difference will cause adifferent kind of quality, that is, luminosity, to appear in addition to the white and a further increase willmerely cause luminosity to become stronger." (Wallach, 1976, p. 8) Thus, the luminosity that occurs

with increments directly contradicts Wallach’s Highest Luminance Rule.

3.3.2. Common factor for decrements and increments: Geometric, not photometric

If the highest luminance rule can be applied only to decrements, what rule could apply to bothdecrements and increments? The common denominator appears to be a geometric factor, not aphotometric factor: it is always the annulus that appears white, both in decrements and in increments. Sofar, anchoring rules have been considered only in terms of luminance relations. This observationsuggested that spatial relations play an important rule.

Wallach’s disk/annulus stimuli lend themselves readily to a figure/ground analysis and Gilchrist andBonato (1995) formulated the hypothesis that lightness values are anchored, not by any specialluminance value, but by the surrounding or background region. The rule, which they called the surroundrule says that for simple displays, lightness is anchored by the surround, which always appears white.Gilchrist and Bonato tested the surround rule against the highest luminance rule using both adisk/annulus configuration and a disk/ganzfeld configuration. The disk/annulus experiment produceddata roughly consistent with the surround rule but with some influence of the highest luminance rule.The disk/ganzfeld data were completely consistent with the surround rule.

4. Area effects

In the disk/ganzfeld experiment, the figure/ground distinction was confounded with relative area. Li andGilchrist (in press) conducted a series of experiments to separate relative area from figure/ground. Tocreate a stimulus pattern that fills the entire visual field, the observer’s head was placed inside a largeacrylic hemisphere. A simple two-part pattern was painted on its opaque interior. Relative area wasvaried while figure/ground arrangements were held constant. Schematic representations of these stimuliare shown in Figure 5, along with the results, given in Munsell values.

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4.1. Surround rule fails

The net result is that the surround rule was completely undermined. In the incremental large ovalcondition, illustrated in Figure 5f, the background appeared middle gray (Munsell 6.0/), not white, as itshould according to the surround rule. What had appeared to be a matter of figure/ground turns out to bea matter of relative area. This is seen clearly in the split dome conditions, shown in Figure 5a and 5b. Asthe darker region increases its area from half the visual field (Figure 5a) to most of the visual field(Figure 5b), its lightness moves from Munsell 4.5/ to 7.8/. This finding brings to mind Helson’s (1964,p. 292) comment that: "...we need assume only that within certain limits area acts like luminance, that is,

increase in area has the same effect as increase in luminance."

4.2. Highest luminance plus area

The highest luminance rule was rescued but combined with a tendency for the largest area to appearwhite. Thus anchoring under minimal conditions (two regions in the visual field) appears to dependupon both photometric and geometric factors. We can say that the larger a surface, the lighter it appears,or "the larger the lighter" for short. But "the larger the lighter" does not apply under all conditions. Itapplies only when there is a conflict between the tendency for the highest luminance to appear white andthe tendency for the largest area to appear white. As long as the highest luminance has the largest area,there is no conflict and that region becomes a very stable anchor. Darker surfaces are seen relative tothat anchor simply according to the ratio principle, and "the larger the lighter" does not apply.

4.3. The Area Rule

By comparing our results with others in the literature we were able to formalize a rule governing theeffects of relative area on perceived lightness that had not been previously recognized. The rule, whichwe call the Area Rule, describes how relative area and relative luminance combine to anchor lightnessperception. The rule is this: In a simple display, when the darker of the two regions has the greaterrelative area, as the darker region grows in area, its lightness value goes up in direct proportion. At thesame time the lighter region first appears white, then a fluorent white and finally, self-luminous.

The term "in direct proportion" here has the following meaning. When the darker region occupies 50%or less of the total area, its perceived lightness is determined simply by its luminance ratio with thelighter region according to Wallach’s ratio principle and the highest luminance rule (the lighter regiontaken to be white). As the area of the darker region grows from 50% to 100% of total area, its lightnessgrows proportionately from this value toward white. We have empirical data supporting this claimregarding the lightness value at 50% and 100%, but we are making an assumption that the transition inlightness is smooth between these values.

Strictly speaking the rule applies to visual fields composed of only two regions of non-zero luminance.Application of the rule to more complex images remains to be studied.

4.3.1. Other empirical findings

There are at least ten other papers in the literature concerning the influence of area on lightness orbrightness. We will see that a great deal of order is brought to this part of the literature when theseresults are analyzed in terms of anchoring in general, and the Area Rule in particular.

Wallach (1948) tested three annulus-to-disk area ratios in his experiments with decremental disks: 4:1,1:1, and 1:4. The first two of these yielded identical results. Relative area had an effect on perceivedlightness only in the third case. When the disk, which was the darker of the two, had an area four timesgreater than that of the annulus, the disk appeared lighter than it would otherwise appear. The Area Rulepredicts no difference between the 4:1 and 1:1 area ratios because the darker region (the disk) does nothave the largest area. But the rule does predict a difference between the 1:1 and 1:4 area ratios becausethe darker region does have the larger area. The darker region is predicted to lighten and this is whatWallach reported.

Using lighting conditions similar to those used by Gelb, Newson (1958) spotlighted a display consistingof a square target surrounded by a brighter square annular region. Holding both center and surroundluminances constant, Newson tested perception of the center square while he varied the area of thesurround from zero to an area roughly equal to that of the center square. This range is just the rangewithin which the Area Rule applies. He obtained a pronounced effect on the lightness of the square.Moreover, his curve (Newson, 1958, Figure 4, p. 94) reaches an asymptote just where the areas of thecenter and surround become equal, suggesting that additional increases in the area of the surround wouldhave no further effect on the lightness of the center.

Kozaki (1963) tested brightness using a haploscopic technique and a square center-surround displayembedded in darkness. Because the area of the surround was always greater than the area of the testfield, the Area Rule would apply under the conditions in which her test field was an increment. Withincrements, she obtained an area effect like that we obtained, consistent with our Area Rule. But she alsoobtained a weak area effect when the test field was a decrement.

Helson and his associates (Helson, 1963, 1964; Helson & Joy 1962; Helson & Rohles, 1959) varied therelative area of either white or black stripes on a gray rectangle in an attempt to resolve the paradox oflightness contrast versus lightness assimilation, as in the classic von Bezold (1874) spreading effect.Their results are not consistent with the Area Rule. We are unable to resolve this discrepancy beyond theobservation that the von Bezold effect may involve a relatively low level kind of space averagedluminance.

Burgh and Grindley (1962) reported no effect of area on perceived lightness using the traditionalsimultaneous lightness contrast display. However, it is crucial to note that they achieved their areachanges by magnifying or minifying the entire display. As a consequence, the relative area between eachgray target and its background was never changed, so this outcome does not contradict the Area Rule.

Yund and Armington (1975) also tested the dependence of brightness on relative area in a disk/annulusdisplay. But, contrary to all the other studies, they tested the effect of the darker region on the brighter,an effect known to be either tiny or nonexistent (Heinemann, 1971; Freeman, 1967, p. 173).

We will not comment on two studies (Diamond, 1962; Whipple, Wallach, & Marshall, 1988) in whicheffects of area were studied because area was confounded with separation between test and inducingfields in those studies.

Four additional studies of brightness and area by Heinemann (1955), Diamond (1955), Stevens (1967),and Stewart (1959) are consistent with the Area Rule. These studies were part of the brightnessinduction literature and will be considered in Section 10 as part of a general review of that literature.

4.4. Luminosity and the Area Rule

4.4.1. Area Rule implies luminosity:

The surround rule had been proposed as a resolution of the apparent contradiction between the highestluminance rule and the perception of luminosity. If the surround rule must be abandoned, can the AreaRule resolve this contradiction and explain both surface lightness perception and luminosity? After all,Li and Gilchrist did obtain luminosity perception in several of their domes experiments even thoughtheir stimuli were nothing more than opaque surfaces. The Area Rule does appear to greatly illuminate

the relationship between opaque surface lightness and self-luminosity. In general we can say thatluminosity perception occurs at one extreme end of the zone to which the Area Rule applies; that is,when a given surface is high in relative luminance but at the same time low in relative area.

4.4.2. Luminosity induction versus grayness induction.

When the luminance difference between two regions of the visual field is increased, two basic outcomes(or some combination) are possible. The darker region might remain perceptually constant while thelighter region moves toward and into a self-luminous appearance. This might be called luminosityinduction; it occurred in Gilchrist and Bonato’s disk/ganzfeld experiments when the disk was anincrement. Alternatively, the lighter region might remain perceptually constant while the darker regionbecomes darker and darker gray. This occurs in disk/annulus experiments (Heinemann, 1955; Wallach,1948) when the disk is a decrement. Heinemann calls this brightness induction, but we will use the termgrayness induction to distinguish it from luminosity induction, and to capture the notion of two opposingdirections, grayness and luminosity.

The question then becomes, when the luminance difference between two regions is increased, whatdetermines whether this increased difference is experienced as an induction of luminosity into thebrighter region or grayness into the darker region? This, of course, is another way of stating theanchoring problem. Indeed Schouten and Blommaert (1995a) have put the problem in this way.

In short the answer appears to lie in relative area. When the darker region is large relative to the lighterregion, most of the effect is expressed as luminosity induction in the lighter region, with only a smallamount of grayness induction. But when the area of the darker region is small relative to the lighterregion, there is very little luminosity induction; most of the effect is expressed as grayness induction inthe darker region.

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Figure 6 represents what is currently our best understanding of the appearance of the two regions in asimple framework as the relative area shifts from the lighter region to the darker. Typically a figure ofthis kind would plot perceived lightness and luminosity as a function of luminance ratio, with relativearea held constant. Note that in this figure we have plotted lightness and perceived luminosity as afunction of relative area, with luminance ratio held constant! We can understand this graph by walkingthrough it, moving from left to right as the x-axis shows increasing relative area of the darker region.Beginning with the dark/light border at the extreme left eccentric position; the darker region is verysmall relative to the lighter region. In this case the lighter region will appear white and the lightness ofthe darker region will depend simply on its luminance ratio with the lighter region. Now, as the bordershifts from the extreme left eccentric position to the center position, no change will occur in theperception of either region because all of these stimuli lie outside the zone of applicability of the Area

Rule. Only when the border crosses the midpoint and begins to move toward the right hand eccentricposition does the Area Rule apply and begin to produce perceptual changes.

As the border passes the midpoint, the darker region begins to grow lighter and lighter. The lighterregion continues to appear white, gradually becoming a more fluorent white (Evans, 1959, 1964, 1974),despite the constant luminance ratio. But the lightening of the darker region is a stronger effect than thebrightening of the lighter region. For example, with a luminance ratio of 30:1, the lightness of the darkerregion can move all the way from black to white (the lightness of the darker region approaches white asits relative area approaches 100%), while the lighter region remains at white. This implies a perceptualcompression. That is, the difference between the perceived lightness of the darker region and theperceived lightness of the lighter region is reduced even though the physical difference remainsconstant. The perceived lightness values seem literally to be squeezed between the tendency of thelighter region to appear white by the highest luminance rule and the tendency of the darker region toappear white because of its preponderant area.

4.4.3. Phenomena related to the Area Rule

Several additional effects are associated with this compression, or squeezing. One seems to be anenhancement of the lighter region, causing it to appear as a kind of pre-luminous super white(Heinemann, 1955; MacLeod, 1947). This twilight zone between white and luminosity has been termedfluorence by Evans (1959, 1974). We believe this may be the same enhancement phenomenon thatHeinemann reported to occur in his test disk when the luminance of the annulus was increased while itsluminance was still below that of the test disk. As the area of the darker, side approaches 100%, itsperceived lightness approaches white, and as this happens, the lighter region is forced to relinquish itswhite appearance (with its opaque quality) and take on the appearance of self-luminosity.

Schouten and Blommaert (1995a, 1995b) have recently reported a phenomenon that they describe as anovel compression mechanism in the luminance-brightness mapping. They call it brightness indention.Using a display that consisted of two disks within a ganzfeld, Schouten and Blommaert found that whenthe disks are both brighter than the ganzfeld, the ganzfeld background does not appear homogeneous; itappears darker in the immediate vicinity of the disks, creating a kind of dark halo around each. Newson(1958, p. 87) described what appears to be the same phenomenon. This phenomenon occurs only in thezone to which our Area Rule applies, when the ganzfeld, with its large area, is darker than both of thedisks. We believe this happens for the same reason as fluorence and the enhancement effect, namelybecause of the competition between the tendency for the ganzfeld to appear white because it has thegreatest area and the tendency for the two disks to appear white because they have the highestluminance. Apparently in this case the conflict is resolved by sacrificing the perceived homogeneity ofthe ganzfeld.

4.4.4. Luminosity threshold and area: Bonato and Gilchrist

Bonato and Gilchrist (1994) studied luminosity thresholds by measuring the luminance value at which atarget surface begins to appear self-luminous. These experiments were subsequently replicated (Bonatoand Gilchrist, in press) with larger targets. This produced higher thresholds. A 17-fold increase in thearea of the target produced a 3-fold increase in the luminance required for luminosity perception. Acorollary result is that as the luminance of the large target was increased, the increasing luminance ratiobetween the target and its background showed up as a darkening of the perceived surface lightness of thebackground. These results are consistent with the Area Rule.

4.4.5. Maximum luminance is not the same as the anchor

In view of the role of relative area in anchoring it seems no longer appropriate to use the terms anchorand highest luminance interchangeably. The anchor in a given framework is the luminance value thatappears white, and because of the area effect, the luminance value that appears white is not necessarilythe highest luminance. The highest luminance is the same thing as the anchor only when the Area Ruledoes not apply.

It should be further noted that the anchor, or luminance value that appears white, need not appear in aframework as an actual surface. For example, if a small white disk is placed in the center of a domepainted black, the disk will appear luminous and the black dome will appear light gray. Perceived whitelies between these two perceived levels but is not represented by a physical surface. Within aframework, the lightness of a target surface is given by the ratio between the target surface luminanceand the luminance of the anchor whether or not the anchor has the highest luminance and whether or notthe anchor luminance is actually represented by a surface.

4.5. Anchoring rules for simple images: A summary

For simple images, the rules of anchoring can be stated very concretely. Except when the Area Rule isengaged, anchoring is straightforward: The brightest region appears white, and the appearance of eachdarker region depends on its relationship to that, according to the formula:

PR = Lt/Lh x 90% (1)

where PR is perceived reflectance, Lt is the luminance of the target, Lh is the highest luminance in theframework, and 90% is the reflectance of white.

For conditions to which the Area Rule applies the formula must be modified. Although greater precisionwill have to come from additional research, the evidence we have as of now, which has been expressedin graphic form in Figure 6, can be summarized algebraically as follows:

PR = (100-Ad)/50 * (Lt/Lh x 90%) + (Ad-50)/50 * (90%) (2)

where Ad is the area of the darker region, as a percentage of the total area in the field. The formulasimply says that if Ad is 50% of the total area, the perceived reflectance of the darker region is just as itis given in Formula (1) above. As Ad approaches 100%, its perceived reflectance approaches 90%.Between these two endpoints there is a smooth transition. The lighter region has no lightness value otherthan white, but as Ad grows, the lighter region takes on additional qualities, first fluorence, and finally,self-luminosity. This qualitative change, not surprisingly, is difficult to capture mathematically.

5. Anchoring in Complex Images: A New Theory

So far the anchoring approach has been applied only to simple retinal images, and here it has proven itseffectiveness in a compelling way. But the ultimate challenge for a lightness theory lies in the kind ofcomplex images we encounter routinely. How can the rules of anchoring by relative luminance and byrelative area be applied to complex images? Obviously, these anchoring rules cannot be applied directlyto complex images. A more plausible approach would be to decompose the image into components, or

sub-images, and then apply our rules of anchoring to each of these components. But here we encounterseveral difficulties. There are a variety of kinds of components into which the image can bedecomposed. We must find the appropriate one. Even if we do, it is not reasonable to expect that eachcomponent sub-image can be treated in total isolation from the rest of the image. Surely there isinteraction among the sub-images and the exact nature of this interaction must be identified.

5.1. What are the relevant components of a complex image?

Both the phenomenologist Katz (1935) and the Gestalt theorists like Koffka (1935) spoke of regions ofthe image they called fields or frameworks. These are regions of common illumination. All the surfaceslying within a shadowed region, for example, would constitute a field. Applying the rules of anchoringwithin such fields of common illumination makes good intuitive sense. But it is not immediately obvioushow the visual system can identify and segregate such fields. Edge classification (Gilchrist, et al, 1983)would be enormously useful here, but spelling out the rules of edge classification presents its ownchallenge to theory. Another approach, though related, would be to divide the image into coplanarregions, as proposed by Gilchrist (1980), but there are pitfalls here as well. What happens, for example,when a shadow falls across half of a set of coplanar regions. Intrinsic images provide yet another kind ofsub-image. As we shall see, however, each of these schemes fails when confronted with the empiricaldata.

5.2. Framework

We propose to define a framework in terms of the Gestalt grouping principles. A framework is a groupof surfaces that belong to each other, more or less. By this definition of framework it is clear thatcomplex images contain multiple frameworks.

These multiple frameworks can be related to each other in several ways, depending on the distributionof grouping factors in the image. Some images are divided into separate but adjacent local frameworks,like a country is divided into provinces. Some images are structured as a nested hierarchy, with severalsuperordinate and subordinate levels. In yet other cases the frameworks intersect one another (seeFigures 25 and 28).

5.3. Local and global frameworks

The largest framework consists of the entire visual field and will be called the global framework.Subordinate frameworks will be called local frameworks. Local frameworks are defined by localgrouping factors, not by distance. There is no fixed degree of proximity within which a group of regionswill be called local.

A target will always be a member of at least two of these frameworks, the global framework, and one ormore local frameworks. In each framework, target lightness is computed according to Formula 1 orFormula 2 (Section 4.5), just as it is computed in simple images. Except by coincidence, the target willhave a different computed lightness when anchored within each of these frameworks.

5.4. Weighting

According to our proposed model, the net lightness predicted for a given target is a weighted average ofits computed lightness values in each of these frameworks, in proportion to the strength of each

framework. Because the grouping factors are graded, as opposed to all-or-none, and because a givenframework can be supported either by a single grouping factor or by several, frameworks can be said tobe stronger or weaker. The strength of a framework also depends strongly on the size of the frameworkand on the number of distinct patches within the framework. A target that belongs to a frameworkcontaining many distinct patches will be anchored strongly by that framework. A target will be morestrongly anchored by a large framework to which it belongs than to a small framework to which itbelongs.

A stimulus configuration that has been frequently studied in lightness perception involves a singlesuperordinate framework that is subdivided into two subordinate frameworks. Katz’s experimentalarrangement composed of adjacent lighted and shadowed fields is one example. Another is thesimultaneous lightness contrast illusion consisting of two gray targets on adjacent black and whitebackgrounds. We can sketch out fairly well the rules of anchoring for images that contain such twolevels of framework, using the more convenient terms local and global even when the entire stimuluspattern does not fill the entire visual field. The following formula predicts the appearance of the target:

PR = Wl(Lt/Lhl * 90%) + (W-1)(Lt/Lhg * 90%) (3)

where Wl is the weight of the local framework, W-1 is the weight of the global framework, Lhl is thehighest luminance in the local framework, and Lhg is the highest luminance in the global framework.When area effects apply, this formula would have to be modified as shown in Formula 2 (Section 4.5).

5.5. Belongingness and grouping factors

The grouping factors produce the perceptual quality of belongingness, or appurtenance as Koffka (1935)called it. A set of coplanar surfaces appear to belong together and thus constitute a framework. A set ofsurfaces moving in the same direction also constitute a framework, based on the principle of commonfate. A group of surfaces lying in shadow constitute a framework as well.

The strongest factor is probably coplanarity, at least when the luminance range is large (Gilchrist, 1980,p. 533). Classic Gestalt grouping factors like proximity, good continuation, common fate, and similarityare also effective. Edge sharpness, T-junctions, and X-junctions (especially when they areratio-invariant) are important factors in belongingness as well, as we shall see. Finally, many empiricalresults appear to require that retinal proximity be treated as a weak but inescapable grouping factor.Grouping factors can segregate as well as integrate.

5.5.1. Importance of T-junctions

The T-junction appears to be a potent grouping factor. In our model, T-junctions function as illustratedin Figure 7. The general principle seems to be that the two "occluded" quadrants (B & C) appear tobelong to each other very strongly while the "occluding" border seems to provide a strong segregativefactor, perceptually segregating the occluding region (A) from the two occluded quadrant regions.Todorovic’ (1997), Ross and Pessoa (in press), and Anderson (1997) have recently emphasized the roleof T-junctions in such illusions, but they have given somewhat different interpretations. Todorovic’speaks of contrast between regions bounded by the same collinear edge, with the basis of contrastunexplained. Ross and Pessoa propose the idea of contrast reduction at context boundaries, signaled insome cases by T-junctions. Anderson emphasizes the role of T-junctions in producing scission.

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5.5.1. Role of luminance gradients

Luminance gradients are held to segregate the luminance values at their opposite ends from each other.If two different but adjacent luminance values are divided by a sharp edge, they belong together stronglyfor anchoring purposes. But if they are separated by a luminance ramp, the same two luminances will beonly weakly anchored by each other.

This may seem backwards. One might argue that when

5.6. Theoretical value of belongingness.

There are several important theoretical advantages to the belongingness definition of a framework. First,it allows us to define frameworks in terms of retinal variables rather than in terms of a perceivedvariable like perceived illumination, avoiding the percept-percept coupling issue. Second, it allows aunified account of both illumination-dependent errors and background-dependent errors (simultaneouscontrast). If a framework were defined as a region of common illumination, as in the usage of Katz(1935) and Koffka (1935), our anchoring analysis would not work for the simultaneous contrast display(the standard textbook version) because there both local frameworks lie in the same field of illumination.Third, the belongingness construction bypasses the problem of edge classification. At the same timegrouping factors must be identified that create the visual experience of a special region of illuminationsuch as a shadows. The segregating role of the penumbra (luminance gradient) is one. Another might becalled luminance similarity. All regions in the shadow share a lowered luminance relative to regionsoutside the shadow.

It is true that we now have a lot of evidence, both empirical and phenomenological, that edges areperceptually classified. And yet, the basis of edge classification has never been fully spelled out. Indeed,our emphasis on belongingness may well provide a new angle from which to attack the classificationproblem, touching, as it does, on the central problem of perceptual structure. Moreover, the anchoringmodel need not deny that humans can classify edges. Rather the model carries the more modestimplication that lightness computation does not depend on edge classification. Edge classification mightdepend on a process that runs in parallel to that of lightness computation.

5.7. The scale normalization effect.

The term scaling refers to the relationship between the range of luminances in the image (or within aframework) and the corresponding range of perceived lightness values. The range of lightness valuescan be either expanded or compressed relative to the range of luminance values. Unless otherwise stated,our model assumes veridical, or 1:1 scaling; the range of lightness values is equal to the range of

luminance values However, we do postulate a scale normalization effect whereby the range of perceivedlightness values in a framework tends to normalize on the luminance range between black and white(30:1). Whenever the luminance range within a framework is greater than 30:1 some compressionoccurs, but whenever the range is less than this, some expansion occurs, with the amount of compressionor expansion proportional to the deviation of the stimulus range from the standard range (30:1).

6. Testing the model: The staircase Gelb effect

Important aspects of the proposed model are illustrated by a series of experiments conducted byCataliotti and Gilchrist (1995; Gilchrist & Cataliotti, 1994), using a display we will call the staircaseGelb effect. A contiguous series of five squares spanning the gray scale from black to white, in roughlyequal steps, was suspended in midair in a laboratory room and brightly illuminated by a homogeneousrectangular patch of light projected by an ellipsoidal theatrical spotlight mounted on the ceiling at adistance of 2.8 M. from the squares. The observer viewed this display with no restrictions from adistance of 4 M and indicated the lightness of each square by selecting a matching chip from a brightlyilluminated Munsell chart housed in a rectangular chamber that rested on a table immediately in front ofthe observer. The entire lab room was normally illuminated by fluorescent lighting but the illuminationon the five squares was thirty times brighter than the ambient level.

The results are shown in Figure 8. The striking aspect in the results is the dramatic compression in therange of perceived grays. Even though the physical stimulus contains the entire gamut of gray shadesfrom black to white, observers perceive only a range of grays from light middle gray to white.

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6.1. Applying the anchoring model

This result makes sense if we apply our anchoring model. We can treat the stimulus squares as membersof two frameworks, one local and one global. The five squares form a local group based on theirproximity, their coplanarity, and most likely their similar luminance values. In addition each is part ofthe global framework that includes the entire visual field. The application of the model is shownschematically in Figure 9. The diagonal line (which will be called the L-line) shows the lightness valuescomputed solely within the local framework, using formula (1) from Section 4.5. The horizontal line(which will be called the G-line) shows the lightness values of the target squares in the globalframework. They reflect the fact that, without the local group of squares, each square by itself wouldappear white in the global framework.

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6.2. Compromise

Notice that the obtained value for each square lies in between its value in the local framework and itsvalue in the global framework. This compromise lies at the heart of our new theory. In this particularcase the compromise is roughly 30% local and 70% global, but we propose that in various othersituations the balance of the compromise might shift in favor of either the local framework or the globalframework, depending upon the relative strength of these two frameworks.

6.3. Weighting factors in the staircase Gelb effect

Gilchrist and Cataliotti (1994) conducted a series of variations on the staircase Gelb experiment to testthe anchoring model by varying a series of factors that might alter the weighting of the local framework.Two of these factors, articulation and field size, were suggested by the early lightness perceptionliterature, especially in the work of David Katz (1935). Two other factors, that we call configuration andinsulation, were uncovered in the course of our investigation.

6.3.1. Configuration.

Apparently the squares constitute a stronger framework if they are arranged in a Mondrian pattern thanif they are simply arranged in a line. This can be seen in Figure 10. Notice that both in the case of fivesquares and in the case of ten squares the obtained data fall closer to the L-line than to the G-line,suggesting that the local framework is stronger with a Mondrian configuration. Several additionalexperiments are needed to further sort out whether the Mondrian configuration produces betterconstancy than the linear configuration because of the greater number of adjacent ratios, because of thescrambling of the luminance staircase, or because of some other factor.

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6.3.2. Articulation.

Three Mondrians, containing 2, 5, and 10 target surfaces respectively, were presented under the samebasic conditions. The results, shown in Figure 11, make it clear that the strength of the local frameworkdepends strongly on the number of squares in the group, even when the luminance range within thegroup is held constant. Our data indicate that the crucial factor is number of different surfaces, not

number of different gray levels, but this needs to be confirmed under a wider range of conditions. Wecall this factor articulation, defined simply as the number of different surfaces in a framework, indeference to Katz (1935). Perhaps this usage fails to capture the richness of Katz’s concept ofarticulation, but our goal is to be more operational than Katz while using his term in order to recognizehis contribution.

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Katz argued that the greater the articulation within an illumination frame of reference, the higher thedegree of lightness constancy. We are modifying Katz’s usage of this concept somewhat. Our proposalis that the higher the degree of articulation in a target’s framework, the more the lightness of the target isanchored solely within that framework.

6.3.3. Field Size.

According to Katz (1935), the degree of lightness constancy within a given field of illumination dependson the size of the field. But as Rock (1975, 1983; Rock & Brosgole, 1964) has shown so often forvarious factors in perception, size can be defined in either retinal terms or in phenomenal terms. Katzbelieved that both of these meanings of size are effective in lightness constancy and hence he offered hisLaws of Field Size. The first law holds that the degree of constancy varies with the retinal size of a field.He supported it with the observation that if one looks though a neutral density filter held at arms lengthand then slowly brings the filter toward the eye, the degree of lightness constancy for surfaces seenwithin the boundaries of the filter increases as the filter comes to occupy a larger proportion of the visualfield.

The second law holds that constancy varies with perceived size. This he demonstrates by keeping thefilter at arms length while he slowly walks backward away from a wall containing various surfaces. Asthe perceived size of the region of wall seen through the filter increases, so does constancy, even thoughretinal size is now held constant.

Although his second demonstration establishes the effectiveness of perceived size with retinal size heldconstant, his first demonstration does not establish retinal size with perceived size held constant. Whenthe filter is drawn closer to the observer’s eye, the total area of the surfaces seen through the filter growsboth in retinal and perceived size.

6.3.4. Perceived size is crucial, not retinal size

In several experiments we have found perceived size to be effective but not retinal size (when perceivedsize is controlled). We repeated the experiment with the five square Mondrian, but used a Mondrian fivetimes larger both in width and height. Results from the small Mondrian and the large Mondrian are

shown in Figure 12 (Gilchrist & Cataliotti, 1994). The increase in size produced a significant darkeningfor only the black square, but there appears to be a trend for the other squares as well. We obtained littleor no difference when we increased size simply by moving the observer closer to the Mondrian, whichincreased the net size of the display with little or no change in its perceived size. Of course, if theobserver were moved so close that the five squares display fills the entire visual field, this wouldconstitute a qualitatively different stimulus and perceived lightness values would change substantially.

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When Bonato and Gilchrist (1994) found higher luminosity thresholds for targets of larger area, theirtargets were larger both in visual angle as well as perceived size. In a follow-up experiment (Bonato &Gilchrist, in press) they tested luminosity thresholds for the disk in a disk/annulus display. They varied,in a controlled way, both the retinal size of the display and its perceived size, finding the luminositythreshold to vary with perceived size but not retinal size.

Bonato and Cataliotti (submitted) showed the importance of phenomenal size in a different way. Theyfound a higher luminosity threshold for a region perceived as ground than for a region perceived asfigure. Although the area of the two regions was the same in the display (shown in Figure 13), thephenomenal area of the ground region is greater because it is perceived to extend behind the figural(face) region. This hidden part of the background can be called its amodal area (Kanizsa, 1979). Aquantitative analysis of the Bonato and Cataliotti results indicates that the functional area of the groundregion (as this bears on the luminosity threshold) includes only some of the area behind the figure, notall of it. This finding is consistent with those of Shimojo & Nakayama (1990), who used an apparentmotion display to determine how far a ground region is perceived (functionally) to extend behind afigure/ground contour.

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6.3.5. Insulation

Gilchrist and Cataliotti (1994) discovered that a white border surrounding the group of squares seems toinsulate it from the influence of the global framework. A border of black paper has no such effect, as canbe seen in Figure 14. Thus, when a region of maximum luminance completely surrounds a group ofspecially lighted surfaces, it seems to dramatically reduce the belongingness between the group and the

remainder of the visual field. The same insulation effect applies to the single square used in the basicGelb effect as well. Cataliotti and Gilchrist (1995) found that when a white square is placed next to theblack Gelb square, the darkening of the black square is less than half what would be commensurate withthe luminance ratio between them. But both they and McCann and Savoy (1991) found that when thewhite region completely surrounds the Gelb square, the darkening effect is highly commensurate withthe luminance ratio between them.

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At this point this insulation effect remains little more than an empirical result; we have no deeperexplanation for it. But the effect is apparently not reducible to local contrast. When the five squares witha white border were compared to a window panes arrangement in which contact between each of thedarker targets and white was maximized, and a concentric arrangement in which the contact wasminimized, no differences were found, as can be seen in Figure 15. the key requirement seems to bemerely that the inner framework is completely enclosed by a white border.

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7. Anchoring and the Pattern of Lightness Errors

We began by noting two weaknesses of the intrinsic image models: the anchoring problem and the errorsproblem. We have quite an ironic situation here. The anchoring model evolved out of an attempt to fillan important gap in the intrinsic image models, namely the lack of anchoring rules. Yet the resultinganchoring model appears to be inconsistent with an intrinsic image model. If our investigation of theanchoring problem has undermined the intrinsic image models, a study of the errors problem promises tobe no more accommodating, given that the intrinsic image models are, at base, models of veridicalperception.

The errors problem is the challenge of explaining the empirical pattern of errors in lightness perception,under a wide range of stimulus conditions. What kind of visual processing could both achieve theimpressive degree of lightness constancy shown by human observers, and at the same time produce justthat pattern of errors they show?

We are aware that the very concept of perceptual error is fraught with philosophical contention. But weintend to sidestep this issue for the moment. At the same time we will give a very concrete definition. Alightness error will be defined as the difference between the actual reflectance of a target surface and thereflectance of the chip selected by the observer from a well-lighted Munsell chart as having the sameapparent lightness as the target.

One cannot find in the lightness literature either a survey of the pattern of errors that have been obtainedin empirical work or a survey of the pattern of errors predicted by the main theories of lightnessperception. A comparison of predicted and obtained errors shows a profound mismatch; many of theimportant errors predicted by theories simply do not occur, and much of the pattern of obtained errorshas remained untouched by lightness theory.

Although the pattern of errors holds the potential to lead us to an adequate theory of lightnessperception, the range of empirical findings on lightness errors is so vast that it is hard to know where tobegin. Our approach will be to organize lightness errors into two broad classes, illumination-dependenterrors and background dependent errors, corresponding to failures in the two basic kinds of constancy. Acrucial test for any theory of lightness errors is to find a single model that can be applied to both of theseclasses of constancy failure. A canonical stimulus configuration will be taken to represent each class; theKatz experimental arrangement and the textbook version of simultaneous lightness contrast.

7.1. Gilchrist (1988): Failures due to classification problem

Gilchrist proposed a common explanation for illumination-dependent errors and background dependenterrors based on edge classification. This approach is problematic because edge classification, at least forcoplanar edges, is all-or-none while constancy failures are graded. The proposal involves a gradedconcept of edge classification. It was based on an experiment in which targets were placed on adjacentbright and dark backgrounds that appeared to differ in reflectance in one condition but illuminance inanother condition. Thus the two conditions allowed for a test of illumination-independent constancy andbackground-independent constancy under identical stimulus conditions, save for changes in the largercontext surrounding the two backgrounds. Veridical performance would have produced ratio matchingfor illumination-independent constancy but luminance matching for background-independent constancy.The actual results fell between these two poles in both cases, but there was a further unique feature ofthe data. The deviation from ratio matching in the illumination-dependent constancy condition wasexactly equal to the deviation from luminance matching in the background-dependent situation, asshown in Figure 16.

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Gilchrist noted that this pattern in the data could be the result of a compromise between two competingprocesses, one process that produces ratio matches and one that produces luminance matches. The firstof these would be invoked if the edge between the backgrounds were classified as an illuminance edge,

the second if it were classified as a reflectance edge. The obtained data are consistent with an incompleteclassification in both cases. The deviation from ratio matching would be predicted if the illuminanceborder dividing adjacent illuminated and shadowed fields were not classified completely as anilluminance border, but classified to some small degree as a reflectance border. Likewise the deviationfrom luminance matching would also be predicted if the border between the adjacent black and whitebackgrounds were not altogether classified as a reflectance edge. This explanation has the appeal ofsymmetry. It is fully consistent with the logic of an invariant relationship between perceived surfacelightness and perceived illumination level, implicit in both the albedo hypothesis (Woodworth, 1938)and the intrinsic image models described in Section 1.7.

7.1.2. Staircase Gelb data inconsistent

The classification failure model does not work, however, for the staircase Gelb effect. Ifillumination-dependent errors are the result of a failure to fully classify the illuminance border betweentwo regions of different illumination it implies that the failures should be equal in magnitude for allsurfaces, regardless of the reflectance of each. For example, if the illuminance step at the border of afield of special illumination is underestimated (and some portion of the step processed as a reflectancestep) every surface within that field will appear erroneously lighter - but all by the same amount.However, in our staircase Gelb effect we found that each of the five squares shows a different degree oferroneous lightening. Thus the classification failure model cannot explain the data we obtained in thestaircase Gelb experiments. Indeed, Bruno (1994) has reported other problems with the classificationfailure model as well.

7.1.3. Anchoring model consistent with Gilchrist (1988) data.

Although the staircase Gelb effect results cannot be explained by Gilchrist’s classification failurehypothesis, the results of Gilchrist’s 1988 experiments can be explained by the anchoring model. Underthose conditions 100% local anchoring would produce ratio matching while 100% global anchoringwould produce luminance matching. The deviations from these two poles would result from acompromise between local and global anchoring, without any reference to edge classification.

Could it be that the solution to the errors problem and the solution to the anchoring problem are one andthe same? First, we must see if the local/global approach can be applied, not merely to our staircase Gelbdata, but to the whole range of illumination-dependent errors.

7.2. Source of Errors

The claim implicit in this model is that errors in lightness perception are caused fundamentally by theprocess of compromise between local and global anchoring. In general, illumination-dependent errorsare attributed to the inappropriate influence of the global (or superordinate) framework whilebackground-dependent errors (as in simultaneous lightness contrast) are attributed to the inappropriateinfluence of the local framework. Any factor that strengthens the local framework relative to the globalwill tend to increase background-dependent errors but decrease illumination-dependent errors.Conversely, any factor that tends to strengthen the global framework at the expense of the local willincrease illumination-dependent errors but decrease background-dependent errors.

8. The Model Applied to Illumination-Dependent Errors

Illumination-independent constancy is the classic form of lightness constancy. The standardexperimental paradigm for studying this kind of constancy was created by Katz (1935). It consists oftwo side by side fields, separated by a screen so that one side is shadowed while the other side isilluminated, as illustrated in Figure 17. A target surface such as a disk is presented within each field andvarious psychophysical methods are used to determine which shade of gray on the shadowed sideappears equal in lightness to which shade of gray on the lighted side. Our staircase Gelb effect is anillumination-independent constancy situation in the sense that the local framework represents theilluminated region and the outer framework represents the shadowed region. This differs from the Katzparadigm however, in that here the lighted region is totally surrounded by the shadowed region. In theKatz paradigm the lighted and shadowed regions are side by side. When applying our model to thisparadigm we would consider the shadowed region to be one local framework, the lighted to be another,and the two together to comprise the global framework.

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To determine how well the anchoring model can be applied to illumination-dependent errors in general,let us survey what is known about these failures in the literature. We will consider five factors that havebeen shown to affect the size of errors.

8.1. Errors associated with level of illumination

The fundamental form of illumination-dependent constancy failures is underconstancy, notoverconstancy (Brunswick ratios under, not over 100%). Underconstancy implies either that surfaces inthe brightly illuminated regions tend to appear lighter gray than they really are or that surfaces inshadowed regions tend to appear darker gray than they really are, or both. This follows from local/globalanchoring. To make this concrete, consider a Katz-type constancy experiment in which a white target inthe shadow has the same luminance as a black target in the lighted field. Complete global anchoring isequivalent to luminance matching, and luminance matching equals zero lightness constancy. Completelocal anchoring would produce 100% lightness constancy, as long as the local framework contains a realwhite. Therefore any local/global compromise can yield only constancy values between zero and 100%.

8.2. Errors associated with reflectance of target.

Perhaps it has been assumed that illumination-dependent constancy failures apply equally to all shadesof gray; that, for example, when surfaces in a lighted region appear lighter than their true value, allshades of gray are lightened by the same degree. However, our anchoring model predicts thatillumination-dependent lightness errors are not uniform for targets of different reflectance within thesame framework. The model predicts a gradient of error: in bright illumination, the lower the targetreflectance, the greater the error. In shadow the reverse is predicted. This gradient of error follows fromthe fact that the luminance range in the global framework is greater than that of any local framework. As

a result the global lightness assignments for a group of surfaces will be compressed relative to their localassignments.

There are few data that can be used to test this prediction because traditional techniques for measuringconstancy allow only a measurement of the overall constancy failure, not the specific failure of aparticular target. For example, in Katz’s original lightness constancy experiment, the task for theobserver was to adjust the gray level of a disk standing in one field of illumination so that it appearsequal in lightness to another disk standing in the neighboring field of illumination. The problem is thatwhen the match is made the experimenter doesn’t know the perceived shade of gray of either disk, onlythat the two disks appear to be the same shade of gray. Testing for a gradient of error requires matchesmade with a Munsell chart.

The data from our staircase Gelb experiment (Cataliotti & Gilchrist, 1995) which were collected using aMunsell chart, show a pronounced gradient of error, as can be seen in Figure 8.

Gilchrist (1980) tested lightness constancy using different depth planes to produce the lighted andshadowed fields and the data were taken using a Munsell chart. Each plane contained two surfaces: onewhite and one black. In the lighted plane, the black showed an erroneous lightening while the whiteshowed no error, while the reverse occurred in the shadowed plane. This is the same gradient of errorpredicted by the model.

8.3. Errors associated with reflectance of backgrounds

Helson (1943) conducted a series of lightness constancy experiments using Katz’s side-by-side fields ofillumination and shadow illustrated in Figure 17. White, gray, and black backgrounds were used.Constancy was best for the white backgrounds (though not great there either), worse for the gray, andpoorest for the black. The same pattern of results was obtained by Kozaki (1963, 1965) as well.Leibowitz, Myers, and Chinetti (1955) tested constancy for a middle gray target on a background thatwas either white, gray, or black. They obtained almost zero constancy, that is luminance matching, forthe gray and black backgrounds, but much better constancy for the white background. This finding ofgreater constancy on backgrounds of higher reflectance has remained simply a curiosity in the literature.It does not flow from any current theories, nor have any explanations been offered, as far as we know.

This finding does flow from the asymmetry inherent in anchoring by white (as opposed to black). Withdark gray backgrounds, each target will usually be the highest luminance in its local framework, and willhave a local assignment of white, regardless of its actual reflectance. Thus the targets will have the samelocal assignments even when they differ substantially in reflectance. In effect, this throws the decision toglobal anchoring, which finds a match when target luminances are equal; that is, zero constancy.

To be concrete, imagine that a light gray target on the shadowed side and a dark gray target on thelighted side have equal luminance values. They will have the same global assignments and, assumingeach is the highest luminance in its local framework, they will have equal local assignments (white).Thus the two targets will be perceived as the same lightness, but this represents zero constancy.

8.3.1. The key: Increments versus decrements

But in fact, the special role of the highest luminance in the new model implies that the reflectance of thebackground is not the crucial variable, per se; but whether the targets are increments or decrements. This

leads to the prediction that a pair of incremental targets on a background of higher reflectance couldshow less constancy than a pair of decremental targets on a background of lower reflectance. By varyingboth the reflectance of the background and the reflectance of the targets, Kozaki (1963, 1965) has shownthat this is indeed the case.

8.4. Errors associated with articulation level

It has been shown that good constancy requires rich articulation and that pronounced failures ofconstancy occur when a field of illumination is poorly articulated.

8.4.1. Burzlaff experiments.

Although there has been virtually no serious discussion of this concept in the past 50 years, there is anample amount of empirical evidence for the effectiveness of the concept of articulation. One can findseveral publications in the early literature that deal with the topic. A good example is the elegant workof Burzlaff (1931). Burzlaff conducted two lightness constancy experiments, one with low articulationand one with high articulation. His first experiment was modeled after one of Katz’s basic techniques forstudying lightness constancy. Observers were presented with two gray disks. One was mounted in aposition relatively near a light source; the other was placed some distance away from the light sourcewhere the illumination level was 20 times lower (see Figure 18). The task of the observer was to adjustthe gray level of one of the two disks to make it appear the same shade of gray as the other disk. LikeKatz before him, Burzlaff obtained very poor lightness constancy under these conditions.

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But in another experiment, Burzlaff presented a cardboard panel containing 48 chips of different shadesof gray spanning the entire black/white scale, arranged in ascending order, in the position near the lightsource, and a second panel containing the identical 48 patches arranged in a haphazard fashion in theposition distant from the light source. This is probably the earliest use of a Mondrian pattern in theliterature.

The task of the observer was to select a patch from the near panel that appeared to be the same shade ofgray as a given target patch in the far panel. Under these conditions Burzlaff obtained excellent lightnessconstancy, with hardly any measurable failure of constancy at all.

Consistent with local and global anchoring, the higher degree of articulation in Burzlaff’s cardboardpanel (as compared with the single disk) made the local frameworks much stronger (or, what isequivalent, increased the belongingness of each surface in the local framework to every other surface inthat framework). Strong local anchoring and a real white in each local framework produces veridicalperception. In the case of the single disk, local anchoring is weak and lightness is strongly determined

by global anchoring. Global anchoring produces luminance matching and, in constancy experiments thismeans poor constancy.

8.4.2. Arend & Goldstein experiments.

Arend and Goldstein (1987; see also Arend & Spehar, 1993a and 1993b) simulated on a CRT screeneither two disk/annulus patterns, one in bright illumination and one in low, or two Mondrian patterns.Observers adjusted the luminance of a target in one illumination to match, for either lightness orbrightness, a target in the other illumination. The data fell closer to ratio matching for decrementaltargets (Section 8.3.1), for the lightness task (Section 10.6.1), and for the Mondrian stimulus (Section6.3.2), but closer to luminance matching for incremental targets, for the brightness task, and for thedisk/annulus stimulus. All of these are consistent with the anchoring model. In these experiments,luminance matching is equivalent to global anchoring while ratio matching is equivalent to localanchoring. The higher articulation level of the 32-patch Mondrian, compared to the 2-patch disk/annuluspattern, produced almost completely local anchoring.

8.4.3. Schirillo experiments.

Schirillo, Reeves, and Arend, (1990) published a replication of one of Gilchrist’s (1980) experiments onlightness and depth perception. Using a CRT screen as a stereoscope they replicated the basic Gilchristfinding but with larger failures of constancy than Gilchrist obtained. In a follow-up paper, Schirillo andArend (1995) investigated the cause of these failures of lightness constancy. They hypothesized thatthese errors are due to local contrast effects caused by non-coplanar surfaces that are nonethelessretinally adjacent to the target surface. They showed that the failures of constancy are indeed greaterwhen the target surface is situated on the border between the differently illuminated planes than whenthe same surface is embedded within the plane. We would attribute this to the factor of retinal proximitymentioned in Section 5.5.

But even for their target on the illumination border, subjected to a local contrast effect, the degree ofconstancy was far higher in the second paper than in the first. The reason for this seems clearly to dowith articulation, as shown in Figure 19. In the first experiment there were only 2 or 3 surfaces in eachplane, while in the second experiment one plane contained a Mondrian composed of 46 surfaces, theother plane a Mondrian of 15 surfaces.

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8.4.4. Application of model to depth and lightness.

In summary, our proposed model would account for the failure of constancy (and the failure of thecoplanar ratio principle) in Gilchrist’s original experiment and in the Schirillo, Arend, and Reeves

experiment in the following way. Because each plane has low articulation, each constitutes a relativelyweak framework and the global relationships intrude substantially to produce failures of constancy. Thewhite surface in the dimly illuminated near plane, for example, would have a global value somewherearound middle gray and this would cause it to appear somewhat darker than a good white, which it did.Likewise, when the same target appears to lie in the far plane, it would be white in the local frameworkbut again roughly middle gray in the global, causing it to appear somewhat lighter than black, which itdid. When the local frameworks are articulated they become much stronger, resisting global anchoring,thus producing greater constancy.

Work by Wishart, Frisby, and Buckley (1995) shows that the strength of the coplanar ratio effect doesvary with its degree of articulation. But because they measured brightness, and not lightness, this workis discussed later together with a class of stimuli that produce background-dependent errors.

Other work showing an effect of articulation has been reported by Cramer (1923), Henneman (1935),and Katona (1929).

8.5. Errors associated with field size

MacLeod (1932), in a constancy experiment modeled after an ingenious Kardos experiment, tested theperceived lightness of a gray disk located in the center of a cast shadow of variable size projected onto awhite wall. As the shadow grew larger the disk appeared lighter gray and thus constancy increased, justas the law of field size claims. According to our anchoring model, the penumbra at the border of theshadow isolates the shadowed region creating a local framework consisting of two regions, the gray diskand its white surround. Anchored within this framework the disk appears gray. But anchored moreglobally, to the brighter white wall outside the shadow the disk would appear much darker. As theshadow gets larger, the local framework that it represents gets stronger and the gray disk becomesanchored more strongly to its local surround, causing it to lighten.

Agostini & Bruno (1996), Henneman (1935), and Katona (1929) have also reported effects of field sizeconsistent with our anchoring model.

9. The Model Applied to Background-Dependent Errors (Simultaneous LightnessContrast)

There is an obvious structural similarity between the simultaneous lightness contrast display and theKatz paradigm with side-by-side lighted and shadowed regions. Each is organized into two side-by-sidesubordinate frameworks, which together compose a single superordinate framework. For continuity inthe use of the terms local and global we begin by applying our model to the case in which the contrastdisplay fills the observer’s entire visual field. But we must keep in mind that more typically (as in atextbook, for example) the contrast display occupies only a part of the visual field. In that case we mustuse the more cumbersome terms subordinate framework and superordinate framework. The globalframework in that case would include both the contrast display and its surrounding context, changes inwhich can have a pronounced effect on the contrast illusion, as Agostini and Bruno (1996) have shown.

According to our anchoring model, there is both an anchoring component and a weaker scalingcomponent in the contrast effect. We begin with the anchoring component that we believe accounts forthe larger part of the illusion.

9.1. Anchoring component

In the global framework, the two targets have the same assigned lightness value because they have thesame luminance and thus they share the same luminance ratio with the global anchor (refer to Figure20), which in this case is the white background on the right side. But in the two local frameworks thesituation is as follows. Because the local framework on the right includes the global anchor (that is, thewhite background) the target on the right stands in the same luminance ratio to the anchor, both locallyand globally. Anchoring per se doesn’t alter this target. The target on the black background, on the otherhand, has a local assignment that is very different from its global assignment. Locally the target is thehighest luminance so that its assignment is white. Thus a compromise between global and localassignments yields a higher value for the target on the black background than for the target on the whitebackground, how much higher depending on the local/global weighting.

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Virtually the same analysis of the simultaneous contrast display was offered in 1987 by McCann: "Ifglobal normalization of the entire field of view were complete, we would expect that observers wouldreport the two gray squares with identical reflectances would have the identical appearance. If localmechanisms were the only consideration, then the gray square in the black surround should mimic theresults found in the Gelb experiment, and should appear a white, since it is the maximum intensity in thelocal area. Observer results give important information about the relative importance of global and localinteractions. The gray square in the black surround is one lightness unit out of nine lighter than the samegray in the white surround. If local spatial calculations were the only consideration, the gray in blackshould appear a 9.0. If global spatial considerations were the only consideration, the gray in blackshould appear a 5.0. The observer matched the gray in black to a 6.0. In other words, the spatialnormalization mechanism is an imperfect global mechanism. Alternatively, it is a local mechanism thatis significantly influenced by information from the entire image." (On the Munsell scale, 9.0 is whiteand 5.0 is middle gray.)

9.2. The scale normalization component

Because each of the local frameworks in the contrast display contains a limited luminance range, wemust expect a small scale normalization effect, or expansion, that, because the anchor is at the top of thelightness scale, is expressed as a darkening of the darker region in each framework. This means that theonly target affected is the one on the white background.

9.3. Predictions of the model

The above analysis leads to at least the following four testable predictions of the model:

9.3.1. The locus of the error.

In our work on simple or single-framework images we found that the anchoring effects are muchstronger than the scaling effects. Thus if our assumption is valid that local frameworks embedded incomplex images function qualitatively like single-framework images, we must also expect that theanchoring component of our model is stronger than the scaling component. Combining this with the factthat the anchoring component produces a lightening of the target on the black background (but not adarkening of the other target) while the scaling component produces a darkening of the target on thewhite background (but no lightening of the other target), we derive the prediction that most of the errorrepresented by the simultaneous contrast effect occurs for the target on the black background.

It is difficult to determine how much the simultaneous contrast illusion involves a lightness distortion ofthe target on the black background and how much the target on the white background. The difficultystems from the lack of a completely valid measure of perceived lightness. Matches made from astandard Munsell chart indicate that the distortion occurs only for the target on the black background, asthe new model predicts. However, the chips in the standard Munsell chart are presented on a whitebackground. Presumably a Munsell chart with chips on a black background would suggest that thedistortion occurs for the target on the white background.

Gilchrist and Economou (unpublished) obtained lightness matches for the simultaneous contrast displayusing a Munsell chart on a black and white checkerboard background so that each chip borders blackand white equally. The illumination on the Munsell chart was equal to that on the contrast display. Theyconducted three variations of the experiment, using a different group of ten subjects in each condition:(1) both contrast display and Munsell chart on paper, (2) both on CRT, and (3) contrast display on CRT,Munsell chart on paper. The results are shown in Figure 21. Consistent with the anchoring model, thetarget on the black background consistently showed the main error, with a smaller error for the othertarget.

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9.3.2. The size of the error.

Our model predicts that the size of the contrast illusion depends on the relative strengths of the local andglobal frameworks. The local frameworks are relatively weak because they have minimal articulation.As for the global framework, we have so far applied the model to the case in which the contrast displayfills the entire visual field, even though there are no published reports of the illusion under theseconditions. More typically the contrast display is presented within a larger complex image. In that casethe strength of the framework that consists of the contrast display itself depends on how strongly thatframework belongs to the larger image and how strongly it is segregated from the larger image. Whenthe display is presented in a textbook, it is perceived to belong to the page of the book and to the table

on which the book is lying. Thus the global, or superordinate, framework is quite strong, so the illusionshould be quite weak. This is just the typical result: the target on the black background appears onlyslightly lighter (by about 3/4 of a Munsell step) than the target on the white background.

However, we would expect the illusion to be strengthened if the entire contrast display is perceptuallysegregated from the global framework. This outcome has been shown to occur by Agostini and Bruno(1996). They showed that a substantially stronger illusion occurs if the contrast display is illuminated bya spotlight so that the bright illumination perceptually segregates it from the less-brightly illuminatedsurrounding region. Under these conditions the illusion is twice as great, 1.5 to 2 Munsell steps, as whenpresented under more typical conditions, such as in a textbook lying on a table within a lighted room. Asthe scope of the spotlight is gradually enlarged to include more of the surrounding region the strength ofthe illusion is gradually reduced to 3/4 of a Munsell step. The same result occurs when a shadow is castthat falls only on the contrast display.

Here we see another example of Katz’s law of field size. As the size of the spotlight increases, greaterconstancy results.

Agostini and Bruno also found that presenting the SLC display on a CRT screen has the same effect aspresenting it within a spotlight; the illusion is approximately doubled. This is an important finding,given the need to integrate the older lightness literature based on paper displays with the more modernwork done primarily on CRT screens. The implication of their finding is that even the perceivedlightness in a region of the visual field the size of a CRT screen depends ultimately on coexisting stimuliin the larger context.

The model predicts that the contrast illusion can be strengthened by increasing the articulation of thelocal frameworks. A high articulation version of simultaneous contrast is presented in Figure 22c. Herethe white background has been replaced with light gray patches and the black background with darkgray patches. The illusion is strengthened even though the two backgrounds now differ less inspace-averaged luminance. Adelson (1996) has reported the same effect.

9.3.3. No contrast effect with double increments

We noted earlier that, according to our model, the source of the contrast effect lies in the localassignments, not the global because they have equal luminance, and thus equal global assignments. Thisimplies that if local assignments were also somehow equal the illusion would be eliminated. Thesimultaneous contrast display with double increments provides just such a case and, as can be seen inFigure 22, it seems to show almost no contrast effect at all (as reported by Gilchrist, 1988) whencompared with the standard display. We can further predict that no illusion will occur when the doubleincrements version is presented within a spotlight, except that now both targets should appear white.These findings have recently been confirmed in our lab.

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It may be noted that the above application of the model to the simultaneous contrast pattern is essentiallyidentical to the application of the model to illumination-independent constancy using black backgroundsoutlined in Section 8.3. Thus the model predicts that the same double increment conditions that producepoor constancy with respect to different illumination levels, produce good constancy with respect todifferent backgrounds.

9.3.4. Segregating effect of luminance ramps

Various manipulations of the contrast display support the idea that luminance ramps exert a segregativerole, in belongingness terms. It has been generally known that if the sharp black/white boundary of thecontrast display is replaced by a gradient, the illusion is strengthened. By segregating the two halves ofthe display we can say either that the superordinate framework is weakened, or its equivalent, that thetwo local frameworks are strengthened. Agostini and Galmonte (1997) have recently shown aparticularly strong version of this illusion.

9.3.5. Belongingness creates the illusion

According to our model, any of the grouping principles should suffice to create a contrast effect. Thiscan be seen in a variety of published reports.

9.4. The Benary effect

The Benary effect (1924), illustrated in Figure 23, was offered by Wertheimer as evidence of the crucialrole of belongingness. The gray triangle that appears to lie on top of the black cross appears lighter thanthe gray triangle that appears to lie on the white background, despite the fact that the two triangles haveequivalent surrounds. Each triangle borders white on its hypotenuse and black on the perpendicularsides. In terms of belongingness, however, when perceptual structure is taken into account, one triangleappears to belong to the black background and it appears lighter than the other triangle that appears tobelong to the white background.

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9.5. White’s illusion.

Strong support for the role of belongingness in contrast effects comes from White’s illusion (White,1981), shown in Figure 24. Here two targets of equal luminance appear different but in a directionopposite to what a contrast model would predict. The gray bars that appear lighter are those that are

surrounded mostly by white while the ones that appear darker are mostly surrounded by black. Yet oneclearly has the phenomenal impression that the lighter gray bars belong to the black stripes, not to thewhite stripes. The main grouping factor at work here is the T-junction.

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9.6. Checkerboard contrast.

The challenge to a lateral inhibition account is pushed to the limit in the phenomenon of checkerboardcontrast, shown in Figure 25 as a variation on a figure presented by DeValois and DeValois (1988, p.229). Here the gray square that appears lighter is completely surrounded by white; the square thatappears darker is completely surrounded by black. In our view this illusion results from the salientgrouping along the diagonals. The square that appears lighter may be completely surrounded by white,but it is perceptually grouped with, and thus anchored by, the black squares.

The effect is rather weak to be sure, and somewhat unstable. But we argue that this is just what shouldbe expected when a fuller accounting of the grouping factors is made. Indeed this stimulus provides agood illustration of multiple local frameworks.

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Each target is a member of three kinds of groups, but each kind of group produces a different effect onlightness. Grouping by rows and columns (based on good continuation) produces no differentiallightness effect because the highest luminance within groups is the same for both targets. Groupingalong the diagonals, based also on good continuation, produces the main effect. Finally our modelalways postulates a weak effect of grouping by retinal proximity. This effect, which is the conventionalsimultaneous contrast effect, opposes the effect produced by diagonal grouping. The weak effect weobserve equals the effect of grouping by diagonals minus the effect of grouping by retinal proximity.

9.7. Agostini and Proffitt: Common fate.

Agostini and Proffitt (1993) demonstrated the importance of belongingness for simultaneous contrast by

eliciting the effect in a novel way. A flock of black disks, a flock of white disks, and two gray diskswere placed randomly on a blue background. Not surprisingly the two gray disks appeared the sameshade of gray. However, when all of the black disks and one of the gray disks were caused to move inone direction, and all of the white disks and one of the gray disks were caused to move in a differentdirection, as shown in Figure 26, a weak simultaneous contrast effect was produced. The gray diskmoving with the black disks appeared lighter than the gray disk moving with the white disks.

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This result shows that common fate causes one gray disk to belong to the group of black disks and theother to belong to the group of white disks. Thus two overlapping frameworks are created and the graydisk that belongs to the framework composed of black disks appears slightly lighter because it is thehighest luminance within that framework. The lateral inhibition account of simultaneous contrast failscompletely here because the two gray target disks have the same blue background.

9.8. Laurinen and Olzak.

Laurinen and Olzak (1994) superimposed shallow sinusoidal modulations on each of the four parts ofthe simultaneous contrast display. They found that the illusion is substantially weakened if in both localframeworks the modulation frequency on the target is different from that of its background. Ourinterpretation of these findings is that the sinusoidal modulation provides the basis for grouping bysimilarity. The difference between each target and its background segregates the target from itsbackground, weakening the local framework, which we regard as responsible for the illusion. At thesame time the similarity of left and right targets increases their belongingness as does the similarity ofleft and right backgrounds, with the effect of strengthening the global (or more correctly, thesuperordinate) framework.

9.9. Wolff illusion

Wolff (1934) created the patterns shown in Figure 27 to illustrate the role of figure/ground segregationin lightness. Each large disk contains equal retinal areas of light and dark. The light gray appears lighterwhen it appears a figure than it does as ground, and the dark gray appears darker as figure than asground. But the background, because it appears to continue behind the figures, has a larger perceivedarea than the figures. Accordingly the pattern on the left comes under the Area Rule and consistent withthe schematic in Figure 6, both light and dark regions should become lighter. This is just what happens.

We can see the local/global compromise at work here as well. The illusion would be much greater ifeach of Wolff’s patterns were painted onto the inside of a dome so that it filled the whole visual field, asFigure 5 confirms. Placing the two patterns on a page, as in Figure 27 creates a superordinate frameworkwithin which there would be no illusion. What we see in Figure 28 is a compromise between no illusion

and the stronger illusion that would occur in the single framework of a dome.

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9.10. Depth manipulations:

Given the importance of coplanarity as a factor in belongingness, one should be able to reduce the sizeof the simultaneous contrast illusion by moving the target squares to a depth plane midway between theobserver and the black and white backgrounds. As long as the black and white regions continue to formthe retinal backgrounds for the two targets, the illusion should be unchanged according to theconventional interpretation based on lateral inhibition. But according to our model this depthmanipulation should decrease the belongingness between each target and its background, weakening thetwo local frameworks, which should in turn weaken the contrast illusion. Wolff (1933) conducted suchexperiments, concluding that (p. 97): "contrast is strongest when both fields lie in the same plane andthere is no contrast at all when the fields are phenomenally situated at a large distance from each other."However, it should be noted that Gibbs and Lawson (1974) found no such effect.

Several investigators have reported that White’s illusion can be reversed using stereopsis to alter thebelongingness relations (Anderson, 1997; Spehar, Gilchrist & Arend, 1995; Taya, Ehrenstein &Cavonius, 1995)

9.11. Adelson’s corrugated Mondrian

Adelson has created a delightful illusion, illustrated in Figure 28, based on the perceived planarity ofsurfaces. Surface B appears lighter than surface A even though they have the same luminance value.According to Adelson (1993, p. 2044), surface A is seen as "a dark gray patch that is brightly lit" whilesurface B is seen as "a light gray patch that is dimly lit." This interpretation of the illusion is consistentwith that of Gilchrist (1980, Gilchrist, Delman, & Jacobsen, 1983), although both Gilchrist and Adelsonhave moved away from this construction.

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This illusion, which is obviously quite strong, follows directly from our anchoring analysis. And again,as in checkerboard contrast, the source of the strength seems to lie in the multiple grouping factors.There are three obvious groupings of the targets: (1) horizontally by rows, (2) vertically by columns, and(3) by local retinal adjacency. Grouping by columns produces no effect (targets share the same groupand the same highest luminance). Grouping by rows produces the main effect. Target B has a higherlocal lightness assignment than target A because the highest luminance in the row is lower for target Bthan for target A. Some of the possible groupings by local retinal adjacency produce a weak effect in thesame direction as grouping by rows. Thus for the corrugated Mondrian, grouping by local adjacencyaugments the main effect of grouping by rows whereas for checkerboard contrast it subtracts from themain effect of grouping by diagonals.

Although Adelson’s results are stronger than simultaneous contrast and checkerboard contrast, they aremuch weaker than those of Gilchrist (1980). This is not surprising from the viewpoint of our model.Adelson presented figures on a CRT screen while Gilchrist used paper 3D displays; both used binocularviewing. Thus in addition to pictorial grouping factors we must also consider grouping factors inherentin stereopsis. In the Gilchrist experiments, stereo factors can be expected to further segregate regions byplanarity while in the Adelson experiments, these same stereo factors work against planarity segregationproduced solely by pictorial cues.

9.11.1. Wishart experiments.

Wishart, Frisby, and Buckley (1995) have shown that the size of the illusion varies as a function of theperceived angle of the folds, as shown on the left side of Figure 29. Gilchrist (1977, 1980) treatedcoplanarity in an all-or-none fashion, but Wishart et al show that the degree of belongingness betweenperceived planes varies with the angle between them.

If we can assume that planarity similarity is a graded variable, then all the main features of Wishart’sdata follow from the new model. (1) As the angle of the fold becomes sharper, the belongingnessbetween the perceived planes weakens, weakening the strength of the superordinate framework (thewhole display). The data shift closer to the local assignments (see Figure 29) increasing the strength ofthe illusion. (2) The change is expressed by target B, not A, because the highest luminance in target A’smain local framework (its row) is the same as the highest luminance in the superordinate framework,thus a change in local/global weighting doesn’t change the predicted lightness of A.

(3) In the flat condition the illusion is weakened because planarity similarity both strengthens thevertical column and weakens the horizontal row. (4) In the vertical condition the illusion is weakenedeven further because the column is further strengthened and the rows are further weakened. The residualillusion in the vertical condition can be attributed to the weak effect of retinal adjacency.

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These same investigators found that the illusion is absent in the simpler display shown in Figure 30. Ifthe basic Adelson effect is based on the perceived difference of illumination in the planes of surface Aand surface B, the illusion should still be present. However, according to our model, the strength of thelocal framework for surface B should drop to almost zero because its degree of articulation has theminimal value of only one. Thus we would expect the percept to be strongly dominated by the globalassignments (A and B equal) and this is what is obtained.

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10. Brightness Induction: Contrast or Anchoring?

During the 1950s and 1960s an extensive literature developed under the rubric of brightness contrast orbrightness induction. The goal of this industry was to explain the perception of brightness and, byextension, surface lightness, with the physiological mechanism of lateral inhibition. These experimentswere very similar to each other in many respects. The stimuli consisted of homogeneous bright patchespresented within a completely dark context. The test display usually consisted of a target square or disk,known as the test field, and an adjacent, nearby, or surrounding region known as the inducing field. Theinducing field was usually brighter than the test field. In addition there was a separate region locatedelsewhere in the visual field, often presented to a separate eye in haploscopic presentation, that served asa matching region and was adjusted by the observer to match the perceived brightness of the test field.Three independent variables were tested repeatedly (1) the luminance, (2) the size, and (3) the proximityof the inducing field. The results of these experiments were interpreted within the theoretical frameworkof contrast (Hering, 1874). But we believe the results make more sense as manifestations of anchoring.The first two variables, luminance and size, can be accounted for by rules of anchoring in simpleimages. The third, proximity, involves the concept of a local/global competition.

10.1. Induction

Perhaps the best known of these experiments is that of Heinemann (1955), who varied the luminance ofan annulus surrounding a disk of fixed luminance. For every setting of the annulus the appearance of thetest field was measured by having the observer adjust a separate, isolated disk so as to match theappearance of the test disk. The basic pattern of data he obtained are shown on the left side of Figure 31.

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There are several features that characterize this curve. First there are two main sections, one roughlyhorizontal and one roughly vertical. The break point in the curve occurs roughly at theincrement/decrement threshold. Heinemann found that as long as the luminance of the annulus is lowerthan that of the disk, annulus luminance has little effect on the appearance of the disk. However, as soonas the annulus luminance surpasses that of the disk, the disk begins to darken.

This basic shape is a manifestation of the highest luminance rule. Figure 31b shows the predicted shapeof the curve under the highest luminance rule. Until annulus luminance supersedes disk luminance,changes in annulus luminance have little effect on the perceived lightness of the disk, which alwaysbeing the highest luminance continues to appear white. Perception of the test disk begins to change onlywhen the annulus becomes highest luminance in the display. From then on, further increases in annulusluminance produce corresponding (that is, consistent with Wallach’s ratio principle) decreases in disklightness.

But when this theoretical curve is superimposed on Heinemann’s empirical curve three discrepancies arerevealed. First, anchoring predicts a drop in the curve with slope -1. Heinemann’s curve drops muchfaster because the comparison disk is presented within a totally dark background. This means that thecomparison disk can never appear as a decrement; it can never be set to a luminance value below that ofthe surround. Whittle (1994a, 1994b) has noted that matching an increment to a decrement is extremelydifficult and subjects will not make such a match if they have the chance to match a decrement with adecrement or an increment with an increment. Further, as Katz (1935) made very clear, although a disksurrounded by an annulus can appear in the surface color mode, a disk surrounded by darkness canappear only in the aperture color mode (as a luminous disk). Matches across these modes are notreliable. As soon as the test field in Heinemann’s experiment becomes a decrement it begins to appeargray and the curve drops precipitously as the observer makes a futile attempt to escape the luminousappearance.

When matching is done with a disk/annulus pattern or a Munsell chart, the slope of -1 is obtained.Wallach (1948), Heinemann (1955) himself, and Gilchrist and Bonato (1995) obtained such a function.

2. Enhancement effect. The first section of the curve is not totally horizontal, as the highest luminancerule demands, but has an upward bulge that has been referred to as the enhancement effect. MacLeod(1947) reported this effect as well.

3. Breakpoint offset. The breakpoint (where the curve drops below its initial y-value) does not alwaysoccur exactly at the increment/decrement threshold (the point where the annulus becomes the highestluminance), as the highest luminance rule predicts. In Heinemann’s study, it occurs a little before thatpoint. In other studies (Heinemann, 1971) it occurs even earlier while in still other studies (Diamond,1953; Horeman, 1963; Leibowitz, Mote, & Thurlow, 1953; Torii & Uemura, 1965) it occurs right at theincrement/decrement threshold.

Two important facts that have already been established provide the key to understanding these featuresof the data. First, the breakpoint offset and the enhancement effect appear and disappear together(Heinemann, 1971; Torii & Uemura, 1965), and second, they both occur only when the Area Rule isapplicable. Heinemann (1971, p. 152) notes: "...as the size of the inducing field is decreased theenhancement effect...becomes smaller." Heinemann and Chase (1995) have recently offered amathematical model that includes an account of the enhancement effect, but neither this model nor theearlier attempt by Diamond (1960) appears to account for why the enhancement effect comes and goeswith relative area.

Both the enhancement effect and the breakpoint offset, as well as their linkage, are readilyunderstandable in terms of the Area Rule. Remember that when the Area Rule is engaged, it produces alightening of the darker (and larger) region, even without a change in the luminance ratio between thetwo regions. This lightening effect has two consequences. First, it puts an "upward pressure" (see Figure6) on the appearance of the brighter region (the disk) causing it to appear as a kind of super white, or theenhancement effect that Heinemann obtained. Second, it causes the annulus to become white evenbefore it becomes the highest luminance. As soon as the annulus becomes the anchor, any furtherincreases in its luminance must make other members of the framework appear to darken. Thus the curverepresenting disk appearance begins its dive even while it is yet an increment, which shows up as thebreakpoint offset.

The enhancement effect and the breakpoint offset have been obtained only in those experiments(Heinemann, 1955; Torii and Uemura, 1965) with inducing fields of larger area than the test field. Whenthe test field is as large as the inducing field (Diamond, 1953; Horeman, 1963; Leibowitz, Mote, &Thurlow, 1953; Torii & Uemura, 1965), neither enhancement nor breakpoint offset occurs.

10.2. Area effects

Besides accounting for Heinemann’s data, the Area Rule is consistent with three other brightnessinduction studies.

Stevens (1967) used a disk/annulus display in a study of area and brightness that was similar to, butmore systematic than Wallach’s study. Diamond (1955) carried out an analogous study except that hisstimulus consisted of a pair of adjacent rectangular regions. Both Stevens and Diamond found apronounced effect of relative area on perceived lightness only when the requirements of the Area Rulewere satisfied; that is, only when the region of lower luminance had the higher relative area. The datapresented by Stevens and Diamond are reproduced in Figure 32. Shading has been added to both graphsto indicate the zone in which the Area Rule applies, showing that effects of relative area onlightness/brightness occur primarily within this zone. Outside of this zone, relative area has little or noeffect on perceived lightness.

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In another variation of the Gelb effect, Stewart (1959) examined how the area or size of a relativelysmall but relatively bright disk affects the perceived brightness of a larger but darker disk upon which itis placed. All of the conditions that Stewart tested fall within the applicable zone of the Area Rule, andhe obtained a pronounced effect of area on brightness, consistent in direction with the Area Rule.

10.3. Separation

The effect of separation between inducing field and test field on the appearance of the darker test field(holding the luminance of each constant) has been studied by Cole and Diamond (1971), Dunn andLeibowitz (1961), Fry and Alpern (1953), and Leibowitz, Mote and Thurlow (1953). All found that theperceived brightness of the darker test field increases with increasing separation between the two. Again,no change occurs in the brighter region. Both McCann and Savoy (1991) and Newson (1958) obtainedanalogous results using similar stimuli and measuring lightness, not brightness. Most of theseinvestigators (Newson and McCann & Savoy are the exceptions) gave a lateral inhibition account oftheir results, according to which the test field becomes brighter with increasing separation due to thewell-known drop off of inhibition strength with spatial separation across the retina.

But we argue that these results are better explained in terms of anchoring. The brighter of the twodoesn’t change in brightness because, whether it is anchored relative to the dark surround or anchoredrelative to the darker target, it has the highest luminance and thus it appears as a constant white. As forthe darker region, the greater its proximity to the brighter region, the more it belongs to the brighterregion and the more its lightness value is determined by its relationship to the brighter region. The lessits proximity to the brighter region, the more its lightness is determined relative to the dark environment,in relation to which its assigned value is white.

There are two lines of research that have attempted to test between contrast and anchoringinterpretations of the separation effect, one using depth separation and one using coplanarity.

10.4. Depth separation: Gogel and Mershon

Rock’s (1975, 1973) classic question, that we applied earlier to the area factor, can be applied to theseparation factor as well. Is brightness determined by retinal proximity or by perceived proximity?Lateral inhibition implies that retinal proximity is crucial while the concept of proximity in three-spaceis more relevant for the perceptual grouping used in our anchoring model. Gogel and Mershon (1969;Lie, 1969; Mershon, 1972; Mershon & Gogel, 1970) showed that when two adjacent surfaces presentedwithin a dark room are separated in stereoscopic depth while maintaining their retinal adjacency, theperceived lightness of the darker region moves toward white just as it does when the two regions areseparated in the frontal plane. This is consistent with Rock’s earlier finding (Rock and Brosgole, 1964)that grouping by proximity depends on perceived proximity, not retinal proximity. And it supports ananchoring account over a lateral inhibition account.

10.5. Gelb effect: Contrast versus anchoring

Many years ago Adhémar Gelb (1929) demonstrated that if a piece of black paper is suspended in midairand illuminated by a beam of light that illuminates only the black paper it will appear white. In this casethe black paper is actually the highest luminance in the scene. But when a piece of real white paper is

brought into the beam and placed next to the original black paper, the black paper becomes perceptuallygray or black. Indeed the black paper is perceptually darkened by any other adjacent paper that has aluminance that exceeds that of the black paper.

Gelb’s marvelous experiment pre-dated the brightness induction literature by several decades. But thesimilarity between Gelb’s stimulus conditions and those of the induction experiments (a few brightly litregions presented in a dark environment) guaranteed that Gelb’s illusion would soon be accounted for interms of contrast based on lateral inhibition. Explicitly contrast accounts of the Gelb effect can be foundin Hurvich and Jameson (1966, p. 107) and Cornsweet (1970, p. 368). Stewart (1959) showed that theamount of darkening of the target depends on both the size and the location of the white inducing regionarguing that this supports the contrast account.

Gelb did not offer an explicit anchoring interpretation of his illusion, but he did clearly reject thecontrast interpretation. Koffka (1935, p. 246) also rejected a contrast interpretation, offering an analysisthat proceeds within the framework of anchoring, although at times he seems to give the lowestluminance equal status with the highest luminance. McCann (1987, p. 279) has been the clearest inattributing the Gelb effect unambiguously to anchoring in general and to the highest luminance rule inparticular: "Here a black piece of paper is the only object in the field of view, by itself it looks a dimwhite, but unquestionably white. When a white paper is put beside the black paper then white lookswhite and black looks black. This is an example of a global normalization process..."

Although both contrast and anchoring interpretations have appeared in the literature, there was noexplicit test between them until recently. Cataliotti and Gilchrist (1995; see also Gilchrist & Cataliotti,1994) exploited another difference between contrast and anchoring treatments of the separation factor.From a contrast perspective, the strength of proximity effects should not depend on whether or not thespace between the test and inducing fields is filled with a coplanar surface. But from an anchoringperspective this can make a crucial difference. For example, moving the position of the highestluminance around within a Mondrian has little effect on lightness values because the various regions ofthe Mondrian are already strongly grouped together by coplanarity and adjacency.

Cataliotti and Gilchrist tested the proximity factor by presenting the Gelb effect in a series ofincremental steps. Their observers were initially presented with a square piece of black paper in thebeam of light. It was duly perceived as white. Then a piece of dark gray paper was introduced into thebeam of light in a position immediately adjacent to the original black paper. The dark gray square wasthen perceived as white because it was now the highest luminance in the display, and the original blacksquare came to appear light gray, relative to the new white. Then a middle gray square was introducedinto the beam of light next to the dark gray square. At this point the middle gray square appeared whiteand both the dark gray square and the black square became darker perceptually. The procedure wasrepeated with a light gray square and finally with a true white square. At this point the stimulusconsisted of a horizontal row of five squares.

The goal was to determine whether the darkening of the existing square or squares by the addition ofeach new and brighter square is greater in magnitude when the new brighter square is adjacent, thanwhen it is in a more remote location. This experiment also served to test a claim made by Reid andShapley (1988; see also Shapley & Reid, 1985) that the effectiveness of edge integration weakens withdistance, much like that of lateral inhibition.

Cataliotti and Gilchrist found that the distance between the highest luminance and each target square did

not have a significant effect on the darkening of the target square. This can be seen in Figure 33 inwhich the perceived lightness value of the original black square is plotted as a function of the maximumluminance in the group. The fact that the obtained data points fall along a straight line indicates that theperceived lightness of the black square depends solely on the luminance ratio between the black squareand the square with the highest luminance, regardless of the spatial distance between them. If thedarkening power of the highest luminance depended on its distance from the black square, the datapoints would fall along a negative slope with a positively accelerated curve.

In another experiment using the entire group of five squares, Cataliotti and Gilchrist found that it makesno statistically significant difference to the perceived lightness of the black square whether the white isplaced immediately adjacent to the black square or placed at the opposite end of the row as in thestaircase order.

These findings imply that the darkening effect of introducing a higher luminance into the group shouldbe considered a matter of anchoring, not contrast.

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10.6. Summary: application of model to reduction conditions.

To summarize, it appears that all of the so-called contrast experiments of the 1950s and 1960s can bemore consistently interpreted as evidence of anchoring rather than contrast. When two homogeneoussurfaces of different luminance values are presented within an otherwise dark void, the degree to whichthey belong together depends strictly on their proximity. When the two are adjacent, they belongtogether strongly, meaning that the lightness of the darker surface is strongly determined by itsrelationship to the lighter surface. The greater the separation between them, the less the darker surfacebelongs to the lighter and the more each belongs to the surrounding darkness. Relative to this darkness,each appears white (and also luminous).

Any additional factor that also increases the degree of belongingness between the two regions willdarken the appearance of the darker region. This can be achieved by: (1) totally surrounding the darkerby the lighter, (2) increasing the perceived size of the two surfaces together, or (3) adding additionalcoplanar and contiguous surfaces. Any factor that reduces the degree of belongingness between the tworegions will lighten the appearance of the darker region. This can be achieved by: (1) reducing theirproximity within the frontal plane, (2) reducing their proximity in depth, or (3) reducing the coplanarityof the two surfaces, that is, changing the angle between their planes while preserving their contiguity.

10.6.1. Lightness versus brightness

The fact that the induction literature can be understood in terms of anchoring is all the more remarkable

given that those experiments studied brightness (to the extent that the distinction was clear at all), andour anchoring model is a model of lightness perception. We see this as evidence that the visual systemattempts to see even such reduced stimuli as surfaces rather than floating light sources. This is not toquestion the lightness/brightness distinction. Rather, we suggest that, under such ambiguous conditions,even when observers attempt to match targets for luminance value, perceived surface lightness intrudes,a bit like the way color names interfere with reporting the color of the ink in the Stroop test. In general,the difference between lightness and brightness instructions can be accounted for simply by changingthe relative weighting of the local and global frames. Brightness instructions shift the weight to theglobal framework while lightness instructions shift the weight to the local framework.

11. Tests of scaling normalization effect

A truncated luminance range was used in both the Mondrian world experiment (Cataliotti & Gilchrist1995) and the split-field dome (Li & Gilchrist, in press), and both produced expansion. This can be seenin Figure 4 by comparing the length of the "perceived" line with the length of the "actual" line.

The scale normalization effect is not large, but it does seem to account for some puzzles. For example, itexplains a small part of the lightness error in simultaneous contrast, in addition to the larger anchoringeffect, as noted in Section 9.2. And it explains why the luminance of a homogeneous surround that isfunctionally equivalent to a checkerboard surround is not exactly equal to the highest luminance in thecheckerboard, but slightly lower than that (Bruno, 1992; Schirillo and Shevell, 1996). And thephenomenon of contrast-contrast (Chubb, Sperling, & Solomon, 1989) concerning the relationshipbetween physical contrast and perceived contrast appears to be reducible to the scale normalizationeffect.

In some cases, however, we have found that what appears to be a scaling problem is really an anchoringproblem in disguise. The compression we obtained in the staircase Gelb effect presented us initially witha serious scaling problem. But according to our anchoring model, this scaling problem reduces to ananchoring problem. All five of the squares are assigned a value of white, or close to white, in the globalframework. Averaging these global assignments into the final compromise necessarily producescompression. Substantial compression effects are also found in a single framework when the Area Ruleapplies, as indicated in Figure 6. Here again the scaling effect reduces to a matter of anchoring. Thecompression is the product of a tension between anchoring by the highest luminance and anchoring bythe largest area.

11.1. Crispening effect

So far we have said nothing about the crispening effect (Semmelroth, 1970; Takasaki, 1966; Whittle,1992), perhaps because we see no obvious way to relate it to our model. But on its face the crispeningeffect is a case of re-scaling; there is expansion near the luminance of the background. Thus it would bereasonable to expect some systematic connection between the crispening effect and our model.

12. Problems for the Intrinsic Image Models

At the beginning of this paper we briefly traced the development of the kind of intrinsic image modelsthat have developed during the past two decades. These models are far more effective than the earliersimplistic and wooden models based on lateral inhibition. They are much more successfully applied to a

wide range of viewing conditions, especially the more complex images typical of our everydayperceptions. They more adequately capture the multi-dimensional character of visual experiencereflected in our perception of both the lightness value of surfaces as well as patterns of illumination onthose surfaces. Intrinsic image models (Adelson and Pentland, 1990; Barrow and Tenenbaum, 1978;Bergström, 1977; Gilchrist, 1979) are well-suited to account for veridicality in surface lightnessperception.

Nevertheless, the intrinsic image models of lightness perception seem to be seriously undermined by aseries of empirical challenges.

12.1. Problem of the staircase Gelb effect data.

Consider how the staircase Gelb display would be treated, according to Gilchrist’s (1979) edgeclassification/edge integration model. In short, the large luminance range within the visual field wouldsignal the need to parse the image into different fields of illumination. This requires identification of theilluminance edges. The basis of edge classification may be ambiguous for many stimuli, but not in thiscase. The sharp, coplanar edges dividing each of the five squares would be classified as reflectanceedges, making it clear that the illuminance border must be the occluding edge forming the outerboundary of the five squares. Given the 30:1 luminance range within this illuminance boundary, the fivesquares would be perceived to range from black to white (under any anchoring rule). This prediction isfar different from the obtained results.

The errors that occur in our staircase Gelb experiment make an especially knotty problem for intrinsicimage models. One can make a provision, such as a small edge classification failure (see Gilchrist,1988) that accounts for the erroneous lightening that occurs in a spotlight. But there is no obvious wayto explain the gradient of error shown by the five squares.

12.2. Problem of area effects.

The finding that perceived lightness depends on the size of a surface does not fit comfortably within theintrinsic image approach. The intrinsic image models are models of veridical perception, based on theconcept of inverse optics. For example, if reflectance and illuminance changes are confounded in theoptics of the retinal image, then they need to be parsed by the visual system in an inverse manner. Butthe area effect cannot be viewed in this way. It is not the case that the reflectance of a surface in thephysical world changes as its size changes. Thus there is no obvious way in which the area effect can beseen to counter an environmental challenge to constancy.

12.3. Problem of articulation effects.

The same argument can be made with respect to the strong effects of articulation that we have found. Ifsurface reflectance is recovered by classifying and integrating edges, there is no reason why the numberof papers within a coplanar group should have such a powerful effect on perceived lightness.

12.4. Problem of errors.

One could say that the intrinsic image models are specialized to account for veridicality while the newmodel is specialized to account for error, but this would be misleading. The intrinsic image models donot account properly for veridicality because they overstate it. On the other hand, the new model would

not be able to make such a precise accounting of lightness errors if it were not, at the same time, able toaccount for veridicality, to the degree that it exists. This is a serious advantage of the anchoring model.

12.5. A response paradox.

For some years now our work has been dogged by a frustrating paradox concerning observer lightnessreports. A piece of gray paper is arranged so that half of it is brightly illuminated while the other half isdimly illuminated. The two halves are divided by a blurred illuminance edge. Observers correctly reportthat it is an illuminance edge and that the surface lightness is the same on both sides of the edge. Yetwhen they make a match from a Munsell chart they always give a higher value to the lighted side than tothe shadowed side. Although the discrepancy can be exacerbated when the task is misunderstood by anaive observer, the paradox runs deeper. Sophisticated observers (including the authors) make the samecontradictory reports.

This paradox is a direct affront to the intrinsic image models, according to which edge classification andperceived surface lightness are tightly coupled. If surface lightness is the product of edge classification,how can such an outcome occur? But if edge classification and surface lightness are treated as parallelprocesses, not serial, as in the new model, such an outcome is possible.

13. New Model vs. Intrinsic Image Models

The model we are proposing has a flavor quite different from the ratiomorphic flavor of the moderncomputational and intrinsic image models. The emphasis on frameworks is characteristic of Gestalttheory, as is the emphasis on global relationships. Belongingness, or "appurtenance," as Koffka called it,is uniquely associated with gestalt theory. Notions of weighted average and compromise that are centralto the new model are alien to the intrinsic image models, suggesting as they do perceptual forces morethan the machine-like precision of edge integration or the all-or-none quality of edge classification.

The functional units into which the image is decomposed seem fundamentally different in the old andnew accounts. Where the intrinsic image models picture an image composed of overlapping layers, ourproposed model pictures an image composed of nested frameworks. And although the lightnesscomputation within these frameworks probably involves both edge extraction and edge integration, thereis no place for edge classification in the new model.

13.1. Classified edge integration

The absence of edge classification in the new model doesn’t mean that the visual system cannot classifyedges (as noted in Section 12.5). But it does mean that the lightness system does not carry out what wecall a classified integration. The process of edge integration originally proposed by Land and McCann(1971), for example, could be called an unclassified integration. That is, the luminance relationshipbetween two widely separated regions in the retinal image is determined by integrating every edge orcontour encountered along an integration path leading from one surface to the other, without regard towhether each edge represents a change in reflectance or in illuminance. But Gilchrist’s (1979) intrinsicimage model specifically asserts that the visual system is able to perform a classified edge integration,an integration of only the reflectance edges between two regions, ignoring or discounting any relativelysharp illuminance borders that occur in the path of integration.

13.2. A Critical Test

Arroyo, Annan, & Gilchrist (1995) have recently conducted an experiment that pits our new modelagainst the earlier model with its classified integration. We were surprised by the results, which appearto decisively favor the new model. The experiment is illustrated in Figure 34.

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13.2.1. Method and results: New model wins.

A large, horizontally rectangular Mondrian containing about 40 gray papers was suspended in midair ina normally-illuminated laboratory room. Choice of gray papers was constrained so that only dark graypapers (between black and middle gray) were permitted on the left half of the Mondrian. The right halfcontained the full range of grays from black to white. A bright rectangular beam of light from anellipsoidal spotlight was projected onto the left half of the Mondrian so that only dark gray papers wereilluminated; no part of any paper lighter than middle gray was allowed to lie within the illumination.

The spotlight was not hidden in any sense. The border of the spotlight that fell across the midline of theMondrian did not coincide with any of the reflectance borders. Indeed the illuminance border containedan obvious penumbra, and the bright illumination was readily seen by all observers.

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But the display constitutes an intriguing critical test, shown in schematic form in Figure 35. Thequestion is, which target patch will appear white? Target patch A, a middle gray square in the spotlightwas not only the highest luminance in its local framework but also the highest luminance in the visualfield, or global framework. Thus, according to our new model, this paper should appear completelywhite. Target B, a white patch in the lower illumination, was the highest luminance, and thus white, inits local framework but light gray in the global framework. Given the relatively high articulation of itslocal framework, B should appear very light gray, almost white.

But according to Gilchrist’s earlier model (1979) the visual system has the ability to perform a classifiedintegration between the gray paper and a white paper on the non-illuminated right side of the Mondrian.An integration of all the edges, except the illuminance edge, along a path from target A to target B, a

white paper outside the spotlight, should reveal that the reflectance value of target B is five times higherthan that of target A. Thus, by the intrinsic image model, the visual system should be able to use thewhite paper as the anchor for the gray paper, even though the luminance of the gray paper is higher.

To summarize the predictions, according to the anchoring model target A should appear fully white andtarget B, slightly off white. According to the intrinsic image approach, target A should appear middlegray, target B, white.

The results we obtained are consistent with the new model. Target A appeared completely white. TargetB appeared white as well, possibly a bit darker than target A, but the difference was not statisticallysignificant.

13.2.2. Low articulation replication.

According to the anchoring model the difference can be increased by weakening the local framework.Annan, et al (1997) repeated the experiment under the minimum conditions of articulation that wouldstill allow a critical test, as illustrated in Figure 36, and obtained just the results predicted. The whitepaper outside the spotlight was perceived as significantly darker than the gray paper in the spotlight.

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13.2.3. Review of 1983 data.

The results of our two critical tests deal a severe blow to the concept of classified edge integration. Whatthen should we conclude about the evidence previously offered in support of this idea?

The data presented by Gilchrist, Delman, and Jacobsen (1983) and by Gilchrist (1988) were far moreconsistent with classified edge integration than with other models current at that time. But those dataturn out to be even more consistent with our new anchoring model. Remember that the intrinsic imagemodel predicts complete constancy, yet substantial failures of constancy appeared in the data. Theanchoring model predicts those constancy failures.

14. Evaluating The Model

How should this model be critically evaluated? First, the model is undoubtedly wrong in some of itsparticulars. It should be regarded as a model still under construction, and we welcome contributions toits further development. Yet there must be a great deal of validity in the basic approach because no othermodel of lightness perception has been able to account for such a wide range of errors.

14.1. Range of application.

The greatest strength of this model is its wide range of applicability. We have surveyed an impressivearray of empirical findings on lightness perception that appear quite consistent with the model.

14.2. Prediction of errors.

Our proposed model is the first model to make an explicit attempt to account for failures of lightnessconstancy in various situations. By comparison, the intrinsic image models predict no errors. Contrastmodels based on lateral inhibition predict many errors, but there is a mismatch between the errors theypredict and the actual errors that occur.

14.3. Unification of constancy failures.

For some years now we have been searching for a single formula that can simultaneously account forboth illumination-dependent constancy failures and background-dependent constancy failures.Historically theories of lightness perception have attempted a unification of illumination-independentconstancy and background-dependent failures of constancy (see Gilchrist, 1994). Several recent attemptshave been made to account for both types of constancy failure. The attempt by Gilchrist (1988) has beenevaluated in Section 7.

Ross and Pessoa (in press) have presented a model in which contrast is selectively reduced at contextboundaries. This approach predicts both the basic illumination-dependent and background-dependentconstancy failures, at least qualitatively, and shows promise for a wider application. But we believe ourproposed model provides the most widely applicable account to date.

14.4. Rigor and concreteness.

Several concepts in this model, like belongingness, framework, and strength of framework, obviouslyrequire greater clarification. Gestalt theory has long been criticized, indeed often dismissed, for a lack ofrigor in its key concepts. To this charge, however, we make two responses. First, rigor should be a goal,not a fetish. Second, the alternative models are not nearly as rigorous as they seem.

We believe that we have shown a strong qualitative relationship between lightness and belongingness,intuitively defined. As in the case of the proverbial drunk, when you lose a quarter you have to look forit in the area where it was lost. There is no point in searching for it somewhere else just because the lightis better. We do not claim to have offered a finished theory; we claim only that we have shown that aserious research program involving concepts of belongingness and anchoring is likely to pay richdividends. Henneman wrote in 1935 (p. 23): "’Articulation of the visual field’ has become recognized asan essential condition for the appearance of ’constancy’ phenomena, though this rather vague term isbadly in need of clearer definition and explanation." But, in the actual event, what articulation got wasnot a better definition, it got forgotten.

The problem of perceptual structure may be difficult but it cannot be avoided. The field of lightnessperception has been plagued with attempts to account for lightness with structure-blind mechanisms. Inthe rush to explain perception at the physiological level as early as possible, there is a reluctance tograsp the nettle of perceptual structure.

This model represents a choice between two broad approaches to theory building. At one pole of thischoice, one can begin with a domain that is sufficiently limited as to allow it to be modeled in relativelyprecise mathematical terms. Then one attempts to broaden the domain to which the model can beapplied, without loss of rigor. At the alternative pole, one takes a wide swath of phenomena and tries tocharacterize, perhaps in fairly broad terms, the nature of these phenomena. Then one attempts to refinethat characterization, without loss of scope. Our model is the product of the latter approach. We aredefending vagueness and imprecision only as a temporary price to pay for progress.

Several models of lightness or brightness (Chubb, Sperling, & Solomon, 1989; Cornsweet, 1970;Grossberg & Todorovic’, 1988; Heinemann & Chase, 1995; Jameson and Hurvich, 1964) are moreformalized than the model we have presented, in the sense that they are couched in mathematicallanguage. Yet none of these models contains a clear anchoring rule, as far as we can see. Thus, if ourargument is accepted that without an anchoring rule (either explicit or implicit in the model) no modelcan predict specific gray shades, one must consider carefully what kinds of rigor are purchased byformalization and what kinds are not.

Neither mathematics nor physiology provides any guarantee of rigor. Mathematical language seems toconfer a cloak of rigor on some models even when no one can specify exactly how the model could befalsified. Lateral inhibition is a concrete concept in that it is based on a physical process. And yet if abright outer annulus is added to the standard disk/annulus display, there is no agreement as to whetherthis should depress the lightness of the disk by adding further inhibition, or enhance it throughdisinhibition.

The anchoring model we offer attempts to predict the specific Munsell values that will be perceived in agiven image. No other theory of lightness perception makes such an attempt, even for simple images.Moreover, we can point to several steps toward greater rigor that we have already taken.

We believe that defining a framework in terms of belongingness does not merely shift the problem fromone of defining framework to one of defining belongingness. Much has been known aboutbelongingness for many years and more recently we have seen a growing accumulation of findings.Koffka (1935, p. 246) emphasizes the importance of coplanarity for belongingness in lightnessperception, and we already have important findings on this topic (Adelson, 1993; Buckley, Frisby, &Freeman, 1994; Gilchrist, 1980; Knill & Kersten, 1991; Schirillo, Reeves, & Arend, 1990). Wishart hasshown that the belongingness between two patches is a graded function of the angle between the planeson which they lie.

Articulation, field size, and T-junctions have been shown to be effective grouping factors across a widerange of displays. Adelson (1993) has produced several provocative new lightness illusions composedlike mosaics, showing that these illusions critically depend on several types of junctions: Y-junctions,Y-junctions, and ratio-invariant X-junctions. A systematic investigation of the role of these junctions inforces of belongingness appears likely to bear fruit. Watanabe and Cavanagh (1993), Todorovic’ (1996),and Anderson (1997) have already made important contributions here as well.

14.5. The role of perceived illumination.

The new model, in its current form, makes no reference to the perception of the illumination, and thismay represent the biggest gap in the model. There is abundant evidence that perceived surface lightnessand perceived illumination are intimately related. The classic hypothesis regarding the relationship

between lightness and perceived illumination is called the albedo hypothesis or lightness-illuminationinvariance hypothesis (Kozaki, 1965; Kozaki & Noguchi, 1976; Logvinenko & Menshikova, 1994;Noguchi & Kozaki, 1985; Oyama, 1968). It holds that there is a unique coupling between perceivedlightness and perceived illumination.

But although the model as it now stands does not specify the role of perceived illumination, there aresome tantalizing findings to suggest a close connection between perceived illumination level and theanchoring of surface lightness. Several investigators have found that the perceived level of illuminationis closely associated with the highest luminance in a framework (Beck, 1959, 1961), or with highestluminance and the largest area (Kozaki, 1973; Noguchi and Masuda, 1971).

14.6. Choice of scaling rule for the global framework.

A serious gap in the new model occurs because the global framework often contains a luminance rangemuch greater than that of the black/white scale. Thus there is no obvious way that lightness valuesshould be assigned within the global framework, although there are several possibilities. (1) The ratioprinciple (1:1 scaling) is the general default rule. But starting with the highest luminance and scaling alllower luminances according to the ratio rule means that all target surfaces more than 30 times darkerthan the maximum will be assigned the same value of black. (2) A projective scaling rule might be used.The entire dynamic range of the global framework can be scaled down proportionately to fit the 30:1range of the black/white scale. Notice that this is equivalent to using a highest luminance rule (=white)and a lowest luminance rule (=black) at the same time. (3) A third possibility, and in our view the mostlikely, is that relative area plays a role in the scaling. For example, a 5:1 range of luminance values thatoccupy very little area within the retinal image might be mapped onto a range of lightness values muchless than 5:1. Only a series of experiments can sort out the appropriate scaling rule.

15. Debt to the Early Literature

The model we are proposing owes a great deal to the early literature in lightness perception. The firstthree decades in this century saw a vigorous development in theory and research on surface lightnessperception. Concepts like field size, articulation, and belongingness that are proving increasinglyessential to an understanding of how surface lightness is processed were standard topics in those days, ascan be revealed by the following list of quotations:

Burzlaff (1931, p. 25): "Katz has often emphasized that for the highest possible degree of colorconstancy it is very important that the field of view appear filled with numerous objects readilydistinguishable from each other... The more homogeneously the visual field is filled, the more thephenomena of colour-constancy recede..."

MacLeod (1932, p32): "When, for instance a section of one’s surroundings is seen as shadowed it is ofconsequence whether or not the shadow occupies a large visual angle and whether the shadowed sectorincludes a variety of objects."

Gelb (1929, p201 in Ellis): "...colour constancy was manifested only when an adequate anduninterrupted survey of the existing illumination was permitted. Anything hindering such a survey (e.g.the reduction screen, minimal observation time, etc.) either destroyed or reduced the phenomenon. Oneof the most important conditions is that the visual angle be large and the field richly articulated."

Katona (1935, p61): "I have found that constancy effects are mainly furthered by enrichment of theperception through better organization, more contours, more form and object-characters, movement,etc."

Henneman (1935, p. 52): "Apparently the more complicated the constitution of the field of observationin the sense of presenting a number of surfaces of different reflectivity, regions of different illumination,and objects set in different planes (tri-dimensionality), the more in accord with its actual albedo-colorwill a test object be perceived."

But the historical continuity was broken by the contrast theorists, as noted in Section 1.4. Consequentlyseveral generations of younger investigators have grown up knowing little of these early ideas. Thesuccess of our proposed model underscores again the importance of citing all the relevant literature. Inpart, our work is an attempt to heal this historical breach.

15.1. Koffka

Overall, our proposed model owes the most to Koffka. In a discussion of color constancy, Koffka (1932)introduces the anchoring problem by invoking the principle of the shift of level which he notes had beenadvanced by Jaensch (1921) as well. Proposing what he calls an invariance theorem, Koffka writes (p.335): "If two parts of the retina are differently stimulated, no constant relationship will exist betweeneach part of the phenomenal field and its local stimulation, but under certain conditions there will be aconstant relationship between the gradient in the phenomenal field and the stimulus difference. I.e. thetwo field parts may, under different conditions, look very differently coloured, but their relation one tothe other or the phenomenal ’gap’ between them will be the same if the stimulus difference is keptconstant. The condition mentioned above is that the two parts of the field belong to the same level." Thisconcept may be easier to understand in reference to Figure 2.

In the same passage, Koffka cites an analogous example concerning orientation. He describes "a publicbuilding on a wide lawn that slants slightly toward the lake." The building appears tilted away from thelake because the perceived horizontal anchor is tilted away from the objective horizontal by the widelawn. Here we see the relevance of the concepts of anchoring and field size for the perception oforientation.

Enormous credit must go to Koffka and the Gestalt theorists for their emphasis on perceptualorganization. In particular Koffka stressed the role of belongingness, which he called appurtenance, asthe key to organization. Indeed the idea, so central to our proposed model, that belongingness is a matterof degree, rather than all-or-none, can be seen clearly in this claim by Koffka (1935, p. 246): "a fieldpart x is determined in its appearance by its ’appurtenance’ to other field parts. The more x belongs tothe field part y, the more will its whiteness be determined by the gradient xy, and the less it belongs tothe part z, the less will its whiteness depend on the gradient xz.." Koffka goes on to identify coplanarityas the most important factor producing belongingness: "Which field parts belong together, and howstrong the degree of this belonging together is, depends upon factors of space organization. Clearly twoparts at the same apparent distance will, ceteris paribus, belong more closely together than field partsorganized in different planes."

Recognition must be given to a more recent Gestalt theorist who worked with Wertheimer early in hercareer. Dorothy Dinnerstein (1965) presented perceptual organization in terms of interacting structures,structures which in turn can be more or less strong. The impact of her ideas can be seen in the model we

have presented.

15.2. Compromise and intelligence

The notion that errors might reflect the resolution of a tension between two poles is not new.Woodworth (1938, p.605) observed that judgments in constancy experiments "usually lie between twoextremes, one conforming to the stimulus and the other conforming to the object". Both the Brunswikratio and the Thouless ratio were created to measure just where the perceptual judgment lies betweenthese two poles. Thouless (1931) spoke of "phenomenal regression to the real object." Rock (1975)attributed constancy failures to unwanted intrusions of the "proximal mode" of perception into the moreobject-oriented "constancy mode." Gilchrist (1988) hypothesized a competition between two processes,one that seeks to equate luminances and one that seeks to equate luminance ratios.

The issue is just how to define the two poles. Each of the above definitions is plagued with either seriouslogical problems or limited range of application. Empirically, however, the local/global polarity seemsto successfully capture the notion of errors as products of a compromise, and to provide a wideapplicability. Sedgwick and Nicholls (1993) have proposed a closely related idea in the spatial domain,viewing spatial errors as the product of cross talk between the information specifying the surface of thepicture and the information specifying the three-dimensional layout of the scene. More precisely, theperception of the three-dimensional layout of the world is influenced by crosstalk from the projectiverelations in the optic array. At the same time, when people are instructed to report on their perception ofprojection relations in the optic array their reports are influenced by crosstalk from informationspecifying the three-dimensional layout of the scene.

Our proposed model presents a picture of visual functioning that, while far more intelligent than contrastmodels based on lateral inhibition, seems far less intelligent than the ratiomorphic approach thatcharacterizes the intrinsic image models. The model implies that surface lightness is not based on acomplete representation of the visual environment. As Adelson (personal communication) puts it:"processing is more powerful than low-level filtering, yet simpler than full 3-D surface description." Theevidence points to a robust system that appears to accept errors even where veridical perception wouldseem to be possible, apparently in exchange for restricting the size of the maximum errors over a widerange of environmental challenges.

15.3. The aperture problem for lightness

Exactly what constitutes intelligent perception may not be clear for many images, even complex images.For example, imagine a richly decorated, brightly illuminated living room. There is an opening in onewall through which you can see into a second, presumably more dimly illuminated room. But you cansee only two adjacent surfaces, call them targets, in the second room. How does your visual systemcompute the perceived gray value for those two targets? The system could anchor the targets using thebright, visually surrounding framework of the first room, by which the they would be seen as two shadesof dark gray. But that computation would produce the correct values only if the illumination in thesecond room were equal to that in the first, and this is not known. Alternatively the system could treatthat piece of the image within the aperture as a world unto itself, anchoring each of the targets relative tothe other, by which they would be seen as white and light gray. This computation would produce thecorrect values only if the highest luminance within the aperture is really a white surface, and this is notknown.

Thus the system is caught between and rock and a hard place. What is the intelligent thing to do?Making a compromise between these two alternative computations is an intelligent response, especiallywhen the weighting in the compromise depends on factors like the number of different surfaces seenthrough the aperture.

In fact most of the images to which we are exposed daily contain the kind of "pockets of ambiguity"exemplified by the aperture lightness problem. Planes, apertures, and regions of special illumination thatcontain less than the full range of gray shades are probably the rule, rather than the exception, even incomplex images. Gibson (1966) argues that ambiguity is characteristic of stimuli presented underreduced laboratory conditions, and that, in ecologically valid optic arrays, the complexity drives out theambiguity. There is truth in Gibson’s observation, but in the case of the hole in the living room wall,increasing the optic complexity in the living room fails to reduce the ambiguity of lightness values forsurfaces seen in the aperture, showing that complexity does not necessarily drive out the ambiguity.Given this reality, there seems to be little option but that a set of rules of anchoring are programmed intothe system that computes surface lightness values.

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Author Notes

The authors wish to acknowledge the support of the National Science Foundation (grant numbersBNS-8909182, DBS-9222104 and SBR 95-14679).

We would like to thank the following people who commented on an earlier draft: Vidal Annan, LarryArend, Nicola Bruno, Elias Economou, Dave Emmerich, Alesandra Galmonte, Walter Gerbino, HolgerKnau, Luiz Pessoa, William Ross, Hal Sedgwick, Dejan Todorovic’, and Paul Whittle.

Requests for reprints should be addressed to Alan Gilchrist, Psychology Dept., Rutgers University,Newark, NJ 07102.

Figure Captions

Figure 1. The ambiguity of luminance ratios.

Figure 2. The anchoring problem.

Figure 3. The scaling problem concerns constancy for the range of perceived grays. Theperceived range of grays can be expanded, compressed, or equal to the range of luminancesin the stimulus.

Figure 4. Evidence for the highest luminance rule, showing the actual range of grays and theperceived range in three separate experiments.

Figure 5. Actual and perceived gray shades in domes experiments (Li & Gilchrist, in press).

Figure 6. Schematic showing how lightness varies as a function of relative area in a two-fielddome with constant luminance values.

Figure 7. Belongingness is increased across the stem of the T, but decreased across the top ofthe T in a T-junction.

Figure 8. Perceived lightness range for five squares in spotlight is compressed relative to theactual range.

Figure 9. Schematic showing theoretical account of compression.

Figure 10. Dependence of compression on configuration of targets. Mondrian configurationproduces stronger local anchoring.

Figure 11. Dependence of compression on number of squares. Greater articulation (moresquares) produces stronger local anchoring.

Figure 12. Dependence of compression on size of display. Larger field size producesstronger local anchoring.

Figure 13. Greater perceived area produces higher luminosity threshold (Bonato & Cataliotti,in preparation).

Figure 14. Insulation: white border produces stronger local anchoring; black border has noeffect.

Figure 15. Test of contrast interpretation of borders effect. Lightness of black target showslittle or no dependence on degree of contact with white border.

Figure 16. Symmetry in illumination-dependent and background-dependent failures ofconstancy.

Figure 17. Katz paradigm for studying illumination-independent lightness constancy.

Figure 18. Burzlaff experiments on lightness constancy and articulation.

Figure 19. Role of articulation in two experiments by Schirillo and colleagues.

Figure 20. Simultaneous lightness contrast display, showing the stronger anchoringcomponent and the weaker scaling component.

Figure 21. Locus of the error in simultaneous lightness contrast. Veridical value is 1.30.

Figure 22. Standard version of simultaneous contrast compared with double incrementsversion and high articulation version.

Figure 23. Benary effect: triangles appear to differ in lightness even though they haveidentical local neighbors.

Figure 24. White’s illusion (White, 1981).

Figure 25. Checkerboard contrast (DeValois & DeValois, 1988).

Figure 26. Schematic of Agostini & Proffitt (1993) experiment. Gray disk moving with blackdisks appears slightly lighter than gray disk moving with white disks.

Figure 27. Wolff illusion (1934). The light and dark areas are equal in each figure. But thebackground regions are perceived to be larger and thus the Area Rule predicts that eachregion in the left hand display will appear lighter than the region of equal luminance in theright hand display.

Figure 28. Adelson’s (1993) corrugated Mondrian. Target B appears lighter than target A.

Figure 29. Application of anchoring model to Wishart experiments. The whole pattern ofresults here is consistent with anchoring if belongingness varies with coplanarity andcoplanarity is treated as a graded variable.

Figure 30. If the illusion is due to perception of target B in a shadowed plane, it should stillbe present. But the anchoring model predicts no illusion here because of the lack ofarticulation in the local framework of B.

Figure 31. Application of highest luminance rule to Heinemann (1955) data.

Figure 32. Application of area rule to data by Stevens and Diamond. Area effects occurmainly under conditions, shown as shaded regions, specified by the Area Rule.

Figure 33. The lightness of the black square in spotlight as lighter squares are added. Thestraight line shows that the darkening of the black square depends on the highest luminancebut not on its distance from the highest luminance.

Figure 34. Stimulus arrangements used to test between anchoring model and intrinsic imagemodel. A square beam of light was projected onto the left half of the Mondrian, whichcontained only dark gray patches. The beam of light was bordered by an obvious penumbra.

Figure 35. Application of intrinsic image model to critical test stimulus.

Figure 36. Low articulation replication of critical test.